#6763919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,23,28)(19,20,29,31)(21,25,24,32)(22,26,30,27);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,30)(22,24)(23,29)(25,27)(26,32)(28,31);(2,6)(3,7)(10,13)(14,16)(17,20,30,32,23,31,22,25)(18,24,26,19,28,21,27,29);(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,22)(20,26)(23,24)(27,31)(28,32)(29,30);(3,7)(5,8)(17,22,23,30)(18,26,28,27)(19,21,29,24)(20,25,31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgh,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=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6671115);