#8115339

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	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8022535);