#8095067

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
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4486 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)(4,6)(5,8)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,2,4,6)(3,5,8,7)(11,17)(15,18)(19,23)(22,24)(25,27,31,28)(26,32,35,33)(29,36,34,38)(30,37,40,39);(1,2)(3,5)(4,6)(7,8)(10,12)(15,18)(16,20)(22,24)(25,27,31,28)(26,33,35,32)(29,38,34,36)(30,37,40,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,27)(26,33,35,32)(29,38,34,36)(30,39,40,37);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,31,34)(26,30,35,40)(27,36,28,38)(32,37,33,39);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,38)(33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002263);