#8087823

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

4427 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)(9,10,15,14)(11,18,17,13)(12,16,22,24)(19,21,20,23)(25,26,31,30)(27,32,29,33)(28,34,37,36)(35,39,38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,20)(14,18)(15,17)(16,21)(19,22)(23,24)(25,27,31,29)(26,32,30,33)(28,35,37,38)(34,39,36,40);(11,17)(13,18)(19,20)(21,23)(25,27,31,29)(26,33,30,32)(28,35,37,38)(34,40,36,39);(9,12)(10,16)(11,19)(13,23)(14,24)(15,22)(17,20)(18,21)(25,28)(26,34)(27,35)(29,38)(30,36)(31,37)(32,39)(33,40);(1,3,5,4)(2,7,6,8)(9,13,15,18)(10,17,14,11)(12,21,22,23)(16,19,24,20)(25,29,31,27)(26,33,30,32)(28,35,37,38)(34,39,36,40);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,18,17,13)(12,16,22,24)(19,23,20,21)(25,30,31,26)(27,33,29,32)(28,36,37,34)(35,40,38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,e^2=gh,f^2=gh,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, 7995019);