#8757570

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

4244 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:

(1,2,5,7)(3,4,8,6)(12,19)(16,20)(18,23)(21,24)(25,26,31,28)(27,34,33,30)(29,32,38,36)(35,40,39,37);(1,3,5,8)(2,6,7,4)(9,10)(11,13)(12,20)(14,15)(16,19)(17,22)(18,24)(21,23)(25,26,31,28)(27,30,33,34)(29,36,38,32)(35,40,39,37);(1,2)(3,4)(5,7)(6,8)(10,15)(11,17)(16,20)(18,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11,14,17)(10,13,15,22)(12,18,19,23)(16,21,20,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3)(2,4)(5,8)(6,7)(9,12,14,19)(10,16,15,20)(11,18,17,23)(13,21,22,24)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,15)(12,21)(14,22)(16,23)(18,20)(19,24)(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=h,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664766);