#8505479

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

4471 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)(3,5)(4,7)(6,8)(9,10,15,12)(11,16,21,19)(13,22,24,17)(14,18,20,23)(25,26)(27,34)(28,33)(29,38)(30,31)(32,37)(35,36)(39,40);(2,3)(6,7)(9,11)(10,16)(12,19)(13,14)(15,21)(17,23)(18,22)(20,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,39)(38,40);(1,2,5,3)(4,6,8,7)(9,10,15,12)(11,19,21,16)(13,17,24,22)(14,18,20,23)(25,28,31,38)(26,33,30,29)(27,35,37,40)(32,39,34,36);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,19,21,16)(13,22,24,17)(14,18,20,23)(25,29)(26,28)(27,36)(30,38)(31,33)(32,35)(34,40)(37,39);(1,4,5,8)(2,6,3,7)(9,13)(10,17)(11,20)(12,22)(14,21)(15,24)(16,18)(19,23)(25,30,31,26)(27,32,37,34)(28,29,38,33)(35,39,40,36);(9,14,15,20)(10,18,12,23)(11,13,21,24)(16,22,19,17)(25,27)(26,34)(28,35)(29,39)(30,32)(31,37)(33,36)(38,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=hi,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, 8412675);