#8371295

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

4565 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8278491);