#8247596

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

4624 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8154792);