#8702653

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

4566 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:

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

Polycyclic presentation:

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

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, 8609849);