#6894236

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6801432);