#10451642

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

4294 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=hi,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, 10358838);