#10360959

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

4352 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)(9,10,15,12)(11,13,16,19)(14,17,23,20)(18,21,22,24)(26,29)(31,35)(32,36)(38,40);(2,6)(4,7)(9,11,15,16)(10,13,12,19)(14,22,23,18)(17,24,20,21)(26,29)(27,33)(32,36)(34,39);(1,3,5,8)(2,7,6,4)(11,16)(13,19)(18,22)(21,24)(25,26,30,29)(27,31,33,35)(28,36,37,32)(34,40,39,38);(1,4)(2,3)(5,7)(6,8)(9,12,15,10)(11,13,16,19)(14,20,23,17)(18,21,22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,13)(10,16)(11,12)(14,21)(15,19)(17,22)(18,20)(23,24)(25,28,30,37)(26,32,29,36)(27,34,33,39)(31,38,35,40);(1,4)(2,3)(5,7)(6,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,23)(16,22)(19,24)(25,29,30,26)(27,35,33,31)(28,36,37,32)(34,40,39,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=ghi,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, 10268155);