#9203672

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

4711 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9110868);