#10320354

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=ghi,e^2=h,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, 10227550);