#8097133

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

4628 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,e^2=h,f^2=h,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, 8004329);