#8096739

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

4701 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,e^2=g,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, 8003935);