#10444920

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

4501 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)(4,6)(5,8)(9,10,14,18)(11,12,16,15)(13,21,22,17)(19,23,24,20)(26,27)(28,35)(29,37)(38,39);(1,2,4,6)(3,5,8,7)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(9,12)(10,16)(11,18)(13,20)(14,15)(17,24)(19,21)(22,23)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=gi,e^2=gi,f^2=h,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, 10352116);