#7526810

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1801 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,27,29,24)(20,25,22,26)(28,32,30,31);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(19,28)(20,22)(24,31)(25,26)(27,32)(29,30);(10,14)(11,15)(18,23)(24,27)(25,26)(31,32);(10,14)(13,16)(17,19,21,29)(18,24,23,27)(20,28,22,30)(25,31,26,32);(1,3,4,8)(2,6,5,7)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434006);