#10424764

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,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, 10331960);