#8932695

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

4542 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)(3,7)(4,5)(6,8)(10,15)(12,17)(16,22)(20,23)(25,26)(27,34)(28,38)(29,30)(31,33)(32,37)(35,36)(39,40);(2,5)(6,7)(9,10)(11,17)(12,19)(13,16)(14,15)(18,23)(20,24)(21,22)(25,27)(26,32)(28,35)(29,39)(30,36)(31,37)(33,34)(38,40);(1,2)(3,6)(4,5)(7,8)(13,21)(16,22)(18,24)(20,23)(25,28,31,29)(26,30,33,38)(27,35,37,39)(32,36,34,40);(1,3,4,8)(2,6,5,7)(9,11,14,19)(10,12,15,17)(13,18,21,24)(16,20,22,23)(25,29,31,28)(26,30,33,38)(27,35,37,39)(32,40,34,36);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,27)(26,32)(28,35)(29,39)(30,36)(31,37)(33,34)(38,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,28)(27,36)(29,33)(31,38)(32,35)(34,39)(37,40)

Polycyclic presentation:

2^9|c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839891);