#8093069

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

6209 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)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,26,21,23)(20,24,28,31)(27,32,29,30);(1,3,4,8)(2,5,6,7)(9,11,12,16)(10,13,14,15)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32);(2,6)(5,7)(11,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32);(2,6)(3,8)(10,14)(13,15)(18,25)(20,28)(23,26)(27,29);(18,25)(23,26)(24,31)(30,32);(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,f^2=ghi,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, 8000265);