#8097182

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

4571 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004378);