#7977567

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

4515 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7884763);