#8077459

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

4443 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)(11,15)(12,16)(13,23)(14,18)(17,21)(19,24)(20,22)(25,26,31,30)(27,28,33,32)(29,37,38,34)(35,39,40,36);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,20)(14,19)(16,17)(18,24)(22,23)(25,27)(26,32)(28,30)(29,35)(31,33)(34,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(11,19)(13,21)(15,24)(17,23)(26,30)(27,33)(29,38)(36,39);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,13)(14,20)(15,23)(17,24)(18,22)(19,21)(25,28,31,32)(26,33,30,27)(29,36,38,39)(34,40,37,35);(1,3,5,4)(2,7,6,8)(9,13)(10,17)(11,20)(12,19)(14,21)(15,16)(18,23)(22,24)(25,29,31,38)(26,34,30,37)(27,35,33,40)(28,36,32,39);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,22)(13,17)(14,18)(16,20)(19,24)(21,23)(25,30,31,26)(27,28,33,32)(29,37,38,34)(35,36,40,39)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=g,f^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, 7984655);