#8092470

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999666);