#8092462

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

5143 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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, 7999658);