#10320358

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

4382 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)(9,10,14,12)(11,16,18,15)(13,17,21,20)(19,23,24,22)(25,26)(27,33)(28,37)(29,31)(30,34)(32,35)(36,38)(39,40);(2,6)(3,7)(11,18)(15,16)(19,24)(22,23)(25,27,32,29)(26,33,35,31)(28,36,30,39)(34,40,37,38);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,28,32,30)(26,34,35,37)(27,36,29,39)(31,38,33,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,29,32,27)(26,31,35,33)(28,36,30,39)(34,40,37,38);(9,12,14,10)(11,16,18,15)(13,17,21,20)(19,22,24,23)(25,30,32,28)(26,34,35,37)(27,36,29,39)(31,40,33,38);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,31)(26,27)(28,38)(29,35)(30,40)(32,33)(34,36)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,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, 10227554);