#8102656

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

4458 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8009852);