#8509673

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
prime radical	#45443	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4526 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8416869);