#10312692

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

4549 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10219888);