#8931016

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

4342 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8838212);