#10247537

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

4301 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154733);