#8092699

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

4670 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:

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

Polycyclic presentation:

2^9|b^2=gi,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, 7999895);