#10368313

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

4478 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=ghi,e^2=h,f^2=gh,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, 10275509);