#501690

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,3,4)(5,6)(7,20)(8,28)(9,34)(10,38)(11,17)(12,18)(13,24)(14,23)(15,35)(16,21)(19,39)(22,46)(25,54)(26,55)(27,62)(29,48)(30,44)(31,65)(32,52)(33,42)(36,63)(37,67)(40,50)(41,68)(43,61)(45,66)(47,58)(49,59)(51,56)(53,60)(57,64);(5,7,21,56,12,27,40,61,11,25,59,30,39,26,10,24)(6,13,38,55,19,44,49,54,17,43,50,62,18,51,16,20)(8,29,57,68,33,35,60,9,23,58,66,37,32,22,31,36)(14,34,53,15,42,41,64,48,28,63,65,46,52,67,45,47);(2,4)(5,8)(6,14)(7,22)(9,30)(10,31)(11,23)(12,33)(13,41)(15,20)(16,45)(17,28)(18,52)(19,42)(21,57)(24,37)(25,35)(26,58)(27,29)(32,39)(34,51)(36,56)(38,53)(40,60)(43,67)(44,63)(46,54)(47,62)(48,55)(49,64)(50,65)(59,66)(61,68);(5,9,21,58,12,37,40,22,11,36,59,29,39,68,10,35)(6,15,38,41,19,48,49,63,17,46,50,67,18,47,16,34)(7,23,56,66,27,32,61,31,25,8,30,57,26,33,24,60)(13,42,55,64,44,28,54,65,43,52,62,45,51,14,20,53)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408886);