#8092760

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

4598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(3,8)(4,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(1,2,5,7)(3,6,8,4)(10,14)(15,20)(16,21)(22,24)(27,33)(30,37)(34,35)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,33)(26,30,32,37)(28,35,36,34)(31,39,40,38);(2,7)(3,8)(10,14)(11,17)(16,21)(18,23)(25,28,29,36)(26,31,32,40)(27,34,33,35)(30,38,37,39);(1,4,5,6)(2,3,7,8)(10,14)(15,20)(16,21)(22,24)(26,32)(30,37)(31,40)(38,39);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=h,f^2=g,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, 7999956);