#8097129

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

4521 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)(27,34)(28,37)(29,36)(30,32)(31,33)(35,39)(38,40);(1,3,5,8)(2,6,7,4)(9,11,14,16)(10,13,17,18)(12,19,22,23)(15,21,20,24)(25,27,31,36)(26,29,33,34)(28,38,30,35)(32,40,37,39);(1,3)(2,4)(5,8)(6,7)(25,28,31,30)(26,32,33,37)(27,35,36,38)(29,39,34,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,23)(17,20)(18,24)(25,29)(26,27)(28,39)(30,40)(31,34)(32,35)(33,36)(37,38);(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,30,31,28)(26,32,33,37)(27,35,36,38)(29,40,34,39);(9,13)(10,16)(11,17)(12,21)(14,18)(15,23)(19,20)(22,24)(25,26)(27,29)(28,32)(30,37)(31,33)(34,36)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,e^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, 8004325);