#407026

order	512
nilpotency class	3
isoclinism	isoclinic class of #407025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14945
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1063 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1063 conjugacy classes of subgroups — too many to draw. Only its 208 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=ef,e^2=i,f^2=i,g^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314222);