#10324514

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

4588 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10231710);