#8092713

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

4537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^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, 7999909);