#7610722

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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 ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(20,27)(24,31)(26,29)(28,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,28)(21,30)(22,25)(24,29)(26,31)(27,32);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19,22,21)(18,23,25,30)(20,29,27,26)(24,32,31,28);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,20,21,29,22,27,19,26)(18,24,23,28,25,31,30,32);(2,8)(6,7)(17,19,22,21)(18,23,25,30)(20,26,27,29)(24,28,31,32)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256838, #72256855, #72256968
maximal normal subgroup of	#72256838, #72256855, #72256968

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7517918);