#8088189

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

4475 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995385);