#8096889

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

4471 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)(3,7)(4,5)(6,8)(9,10,15,13)(11,18,16,12)(14,17,22,21)(19,24,23,20)(25,26,30,34)(27,33,31,28)(29,37,39,32)(35,36,38,40);(2,4)(6,7)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23)(25,27)(26,28)(29,38)(30,31)(32,40)(33,34)(35,39)(36,37);(1,3)(2,6)(4,7)(5,8)(25,28)(26,31)(27,34)(29,36)(30,33)(32,38)(35,37)(39,40);(1,4,5,2)(3,7,8,6)(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,29,30,39)(26,32,34,37)(27,35,31,38)(28,36,33,40);(9,13,15,10)(11,18,16,12)(14,17,22,21)(19,20,23,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,38)(39,40);(9,14,15,22)(10,17,13,21)(11,19,16,23)(12,20,18,24)(25,27)(26,33)(28,34)(29,35)(30,31)(32,40)(36,37)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,e^2=gh,f^2=gh,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, 8004085);