#8931453

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

4434 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)(25,26,31,28)(27,34,33,30)(29,32,38,36)(35,40,39,37);(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)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,5,8)(2,6,4,7)(10,14)(11,17)(16,19)(18,20)(26,28)(30,34)(32,36)(37,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,31,26)(27,34,33,30)(29,32,38,36)(35,37,39,40);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=h,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, 8838649);