#7511903

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,24,27,30)(20,28,22,25)(26,32,29,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,26)(20,22)(24,31)(27,29)(30,32);(10,14)(13,15)(17,19,21,27)(18,24,23,30)(20,26,22,29)(25,31,28,32);(2,7)(3,8)(11,16)(13,15)(17,20,21,22)(18,25,23,28)(19,26,27,29)(24,31,30,32);(1,4,5,6)(2,3,7,8)(18,23)(24,30)(25,28)(31,32)

Polycyclic presentation:

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

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, 7419099);