#7917052

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

4360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7824248);