#7954596

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

4640 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)(11,13)(12,15)(14,17)(16,18)(19,21)(20,23)(22,24)(28,36)(33,38)(35,37)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(1,3,5,4)(2,7,6,8)(10,12)(11,16)(17,20)(19,24)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,4,5,3)(2,8,6,7)(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,29,31,27)(26,32,30,34)(28,37,36,35)(33,39,38,40);(9,14,15,23)(10,17,12,20)(11,19,16,24)(13,21,18,22)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gi,e^2=gi,f^2=i,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, 7861792);