#6403166

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6310362);