#10451624

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4353 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24)(25,26,30,34)(27,29,37,33)(28,32,39,31)(35,40,36,38);(2,5)(6,7)(9,10,14,18)(11,15,16,12)(13,17,22,24)(19,23,21,20)(25,27)(26,29)(28,36)(30,37)(31,40)(32,38)(33,34)(35,39);(1,2)(3,6)(4,5)(7,8)(25,28)(26,31)(27,35)(29,38)(30,39)(32,34)(33,40)(36,37);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,18)(13,21)(14,16)(17,20)(19,22)(23,24)(25,28)(26,32)(27,36)(29,38)(30,39)(31,34)(33,40)(35,37);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19)(25,29,30,33)(26,27,34,37)(28,38,39,40)(31,35,32,36);(9,13,14,22)(10,17,18,24)(11,19,16,21)(12,20,15,23)(25,27)(26,33)(28,35)(29,34)(30,37)(31,40)(32,38)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=hi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358820);