#6763560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	#23976	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,29)(21,24)(22,25)(26,27)(28,32)(30,31);(2,6)(3,7)(10,13)(14,16)(17,19,21,27,22,28,20,30)(18,23,24,26,25,32,29,31);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(3,7)(5,8)(19,28)(23,32)(26,31)(27,30)

Polycyclic presentation:

2^9|c^2=fgh,d^2=fi,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^c=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, 6670756);