#10420148

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

4397 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26)(27,32)(28,37)(29,33)(30,36)(31,34)(35,39)(38,40);(2,5)(6,7)(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20)(25,27)(26,32)(28,38)(29,35)(30,34)(31,36)(33,39)(37,40);(1,2)(3,6)(4,5)(7,8)(25,28,31,29)(26,33,34,37)(27,35,36,38)(30,39,32,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,29,31,28)(26,33,34,37)(27,35,36,38)(30,40,32,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,30)(26,27)(28,39)(29,40)(31,32)(33,35)(34,36)(37,38);(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,24)(19,23)(25,27)(26,30)(28,35)(29,38)(31,36)(32,34)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,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, 10327344);