#8422102

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

4416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8329298);