#8751829

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

5036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8659025);