#6910176

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6817372);