#7623517

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623305
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287303, #72287309, #72287359, …
maximal normal subgroup of	#72287303, #72287309, #72287359, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530713);