#6605642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,22,25,20,29)(19,26,27,31,28,23,30,32);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,29)(22,25)(26,28)(27,31)(30,32);(2,6)(5,8)(10,14)(11,15)(17,19)(18,23)(20,30)(21,27)(22,28)(24,32)(25,26)(29,31);(9,11)(10,13)(12,15)(14,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(11,15)(13,16)(19,28)(23,26)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308774, #72308792
maximal normal subgroup of	#72308774, #72308792

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512838);