#8094980

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

4504 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002176);