#8254681

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

4395 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gh,e^2=hi,f^2=i,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, 8161877);