#8096815

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

4609 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,e^2=i,f^2=i,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, 8004011);