#6711848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1559 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,24)(22,26)(23,25)(27,29)(28,30)(31,32);(10,13)(11,15)(17,18)(19,27)(20,29)(21,24)(22,32)(23,28)(25,30)(26,31);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,19,21,25)(18,23,24,20)(22,29,31,28)(26,30,32,27);(10,13)(14,16)(18,24)(19,25)(26,32)(28,29);(1,3,4,7)(2,6,5,8)(17,20,21,23)(18,25,24,19)(22,27,31,30)(26,28,32,29)

Polycyclic presentation:

2^9|c^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6619044);