#6763790

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,24)(20,30)(21,25)(22,32)(23,26)(27,29)(28,31);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(17,19,23,28)(18,24,26,31)(20,25,32,27)(21,22,29,30);(2,6)(3,7)(10,13)(14,16)(17,20,30,18,23,32,22,26)(19,27,21,31,28,25,29,24);(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,22)(20,31)(23,29)(24,32)(26,27)(28,30);(3,7)(5,8)(19,28)(21,29)(24,31)(25,27)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgh,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6670986);