#7557594

order	512
nilpotency class	3
isoclinism	isoclinic class of #7513980
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

1383 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7464790);