#8751086

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

4579 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658282);