#6908571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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	#37649	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 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)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,30)(21,24)(22,25)(23,29)(26,31)(28,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,28,22,29,20,31)(18,23,24,26,25,27,30,32);(2,6)(5,7)(11,16)(13,15)(17,20,22,21)(18,24,25,30)(19,28,29,31)(23,32,27,26);(10,14)(13,15)(19,29)(23,27)(26,32)(28,31)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^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, 6815767);