#8093087

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

4926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^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, 8000283);