#7507777

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1954 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281985, #72282201, #72282311
maximal normal subgroup of	#72281985, #72282201, #72282311

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414973);