#392721

order	512
nilpotency class	3
isoclinism	isoclinic class of #392719
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14694
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3607 conjugacy classes of subgroups; only the 416 normal subgroups are shown.

normal subgroups only

This group has 3607 conjugacy classes of subgroups — too many to draw. Only its 416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299917);