#7528234

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(20,28)(24,31)(27,30)(29,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,26)(19,25,21,23)(20,29,28,32)(24,30,31,27);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,22,21)(18,23,26,25)(20,27,28,30)(24,29,31,32);(10,14)(13,16)(17,20,22,28)(18,24,26,31)(19,27,21,30)(23,29,25,32);(1,4,5,3)(2,7,6,8)(17,19,22,21)(18,25,26,23)(20,27,28,30)(24,32,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7435430);