#7623150

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=fh,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gh,h^a=hi,g^b=gi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215833
maximal normal subgroup of	#72215833
maximal quotient of	#72226795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530346);