#10349118

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

4605 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=gh,e^2=gh,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, 10256314);