#8097819

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

4621 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)(3,4)(5,6)(7,8)(12,20)(15,22)(18,24)(21,23)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,6)(3,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(28,37)(32,36)(34,40)(38,39);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8005015);