#10412047

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

4399 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10319243);