#7603119

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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,23)(21,31)(22,26)(24,30)(25,29)(28,32);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(19,28)(20,30)(21,29)(23,24)(25,31)(27,32);(10,14)(11,16)(17,19,29,28,22,30,21,20)(18,23,31,24,26,32,25,27);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,29,19,22,28,21,30)(18,24,31,32,26,27,25,23);(2,5)(6,8)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=fgi,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7510315);