#8271926

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

4619 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=i,e^2=hi,f^2=hi,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, 8179122);