#6868058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6775254);