#8425354

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

4607 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,e^2=g,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, 8332550);