#7954619

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

4258 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)(11,19)(12,22)(13,24)(14,18)(15,21)(16,23)(17,20)(27,29)(32,34)(35,36)(39,40);(1,2,5,6)(3,7,4,8)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,17,22,24)(25,26,31,30)(27,32,29,34)(28,33,38,37)(35,39,36,40);(1,3,5,4)(2,7,6,8)(9,12,14,23)(10,16,18,22)(11,13,21,20)(15,24,19,17)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,12)(10,16)(11,20)(13,21)(14,23)(15,17)(18,22)(19,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,22,19,16)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,24)(14,18)(17,20)(19,21)(22,23)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=hi,e^2=h,f^2=gh,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, 7861815);