#10060939

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

4777 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,e^2=hi,f^2=gh,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, 9968135);