#10020840

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

4619 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)(11,17)(15,18)(19,23)(22,24)(25,26,30,33)(27,31,36,39)(28,29,37,32)(34,35,40,38);(2,3)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,36)(26,31,33,39)(28,34,37,40)(29,35,32,38);(2,3)(6,7)(10,14)(15,18)(16,20)(22,24)(25,28,30,37)(26,32,33,29)(27,34,36,40)(31,38,39,35);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28)(26,29)(27,34)(30,37)(31,35)(32,33)(36,40)(38,39);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(1,4,5,8)(2,6,3,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,28)(27,35)(30,32)(31,34)(33,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9928036);