#6614926

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257587, #72257608, #72264263
maximal normal subgroup of	#72257587, #72257608, #72264263
maximal quotient of	#72264751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6522122);