#8199211

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

4448 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:

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

Polycyclic presentation:

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

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, 8106407);