#7930498

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	#39848	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(27,34)(28,31)(35,39)(36,37);(1,2)(3,6)(4,5)(7,8)(25,26,30,33)(27,31,34,28)(29,32,38,40)(35,37,39,36);(10,14)(15,17)(16,21)(22,23)(25,27,30,34)(26,28,33,31)(29,35,38,39)(32,36,40,37);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,37)(36,40)(38,39);(1,3,4,7)(2,6,5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,31)(26,27,33,34)(29,37,38,36)(32,39,40,35);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,32,33,40)(27,35,34,39)(28,36,31,37)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7837694);