#7604016

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518936
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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)(3,7,8,6)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,30)(21,23)(22,31)(24,29)(26,27)(28,32);(2,5)(6,7)(9,11)(10,14)(12,15)(13,16)(18,23)(19,20)(22,27)(24,25)(28,29)(30,32);(9,12,13,10)(11,15,16,14)(17,19,27,20,21,28,22,29)(18,24,31,32,23,30,26,25);(11,16)(14,15)(17,20,22,19,21,29,27,28)(18,25,26,30,23,32,31,24);(1,3,4,8)(2,6,5,7)(18,23)(24,30)(25,32)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511212);