#7584427

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,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, 7491623);