#8360948

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

4584 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)(4,6,8,7)(9,10,15,12)(11,16,18,14)(13,17,23,20)(19,24,21,22)(26,31)(27,28)(33,38)(34,36);(2,3)(6,7)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,22,20,24)(25,26,30,31)(27,32,28,35)(29,33,37,38)(34,39,36,40);(4,8)(6,7)(11,18)(14,16)(19,21)(22,24)(25,27,30,28)(26,32,31,35)(29,36,37,34)(33,40,38,39);(2,3)(6,7)(9,12,15,10)(11,14,18,16)(13,17,23,20)(19,24,21,22)(25,28,30,27)(26,32,31,35)(29,36,37,34)(33,39,38,40);(1,3,5,2)(4,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,21)(23,24)(25,29,30,37)(26,33,31,38)(27,34,28,36)(32,39,35,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8268144);