#6819621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1392 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)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,30)(28,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(20,30)(22,29)(24,31)(25,27);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,29,20,21,30,22,28)(18,24,32,31,23,27,26,25);(10,14)(13,15)(18,23)(24,27)(25,31)(26,32);(17,20,29,30,21,28,22,19)(18,25,32,24,23,31,26,27)

Polycyclic presentation:

2^9|b^2=fg,c^2=gh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313880, #72313904, #72313939
maximal normal subgroup of	#72313880, #72313904, #72313939
maximal quotient of	#72320024

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6726817);