#8093362

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4266 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000558);