#7596262

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
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(11,15)(12,13)(14,16)(17,18,24,19)(20,29,32,26)(21,25,31,28)(22,27,23,30);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,24,18)(20,26,32,29)(21,28,31,25)(22,30,23,27);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,23,21,24,32,22,31)(18,25,27,29,19,28,30,26);(1,4,5,3)(2,6,8,7)(9,11,12,16)(10,14,13,15)(17,21,23,32,24,31,22,20)(18,26,27,25,19,29,30,28);(2,8)(6,7)(17,22,24,23)(18,27,19,30)(20,31,32,21)(25,29,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308774, #72308819
maximal normal subgroup of	#72308774, #72308819

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7503458);