#6810409

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,26)(14,19)(15,20)(17,28)(18,29)(21,36)(22,35)(23,31)(25,40)(27,33)(30,39)(32,38)(34,37);(1,3,4,8)(2,5,6,7)(9,11,15,23)(10,16,20,31)(12,14,22,29)(13,25,28,38)(17,32,26,40)(18,24,19,35)(21,37,39,27)(30,33,36,34);(3,8)(5,7)(9,12,14,23,15,22,29,11)(10,16,19,24,20,31,18,35)(13,25,27,39,28,38,37,21)(17,30,34,32,26,36,33,40);(2,6)(5,7);(9,13,14,27,15,28,29,37)(10,17,19,34,20,26,18,33)(11,21,12,25,23,39,22,38)(16,30,24,32,31,36,35,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717605);