#8504839

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

4551 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,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, 8412035);