#10422428

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

4668 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,e^2=h,f^2=gh,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, 10329624);