#7533312

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,22)(21,25)(23,29)(24,32)(27,28)(30,31);(2,5)(6,7)(11,16)(13,15)(17,19,21,29)(18,23,25,26)(20,31,32,27)(22,28,24,30);(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,32)(22,25)(23,28)(26,30)(29,31);(10,14)(13,15)(18,25)(19,28)(22,24)(23,27)(26,31)(29,30);(1,3,4,8)(2,6,5,7)(18,25)(20,32)(23,26)(27,31)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281922
maximal normal subgroup of	#72281922

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440508);