#393898

order	512
nilpotency class	3
isoclinism	isoclinic class of #393897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14715
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3870 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 3870 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15)(4,20)(5,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,29)(17,45)(19,39)(21,51)(22,35)(24,34)(25,37)(26,36)(27,38)(28,31)(30,46)(32,44)(33,53)(40,43)(41,47)(42,55)(48,56)(49,57)(50,54)(52,59)(58,62)(60,63)(61,64);(1,3)(2,9)(4,16)(5,17)(6,28)(7,18)(8,19)(10,32)(11,30)(12,39)(13,15)(14,31)(20,40)(21,42)(22,48)(23,43)(24,44)(25,49)(26,46)(27,47)(29,35)(33,58)(34,56)(36,41)(37,45)(38,57)(50,60)(51,64)(52,63)(53,61)(54,62)(55,59);(1,4)(2,10)(3,16)(5,21)(6,23)(7,22)(8,24)(9,29)(11,33)(12,20)(13,34)(14,35)(15,40)(17,42)(18,48)(19,44)(25,53)(26,52)(27,54)(28,43)(30,55)(31,32)(36,51)(37,50)(38,59)(39,56)(41,60)(45,64)(46,63)(47,62)(49,61)(57,58);(1,5,8,27)(2,11,14,38)(3,17,19,47)(4,21,24,54)(6,26,7,25)(9,30,31,57)(10,33,35,59)(12,37,13,36)(15,41,39,45)(16,42,44,62)(18,49,28,46)(20,50,34,51)(22,53,23,52)(29,55,32,58)(40,60,56,64)(43,63,48,61)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303185, #72303351, #72303359
maximal normal subgroup of	#72303185, #72303351, #72303359

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301094);