#7973956

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
prime radical	#43000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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)(11,14)(12,15)(13,20)(16,18)(17,21)(19,22)(23,24)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,14)(13,19)(15,18)(17,22)(20,23)(21,24)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(1,3,5,4)(2,7,6,8)(10,12)(11,18)(13,21)(22,23)(25,27,29,34)(26,30,32,38)(28,33,36,35)(31,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,14,18,16)(13,20,21,17)(19,23,24,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(1,4,5,3)(2,8,6,7)(9,13,15,21)(10,17,12,20)(11,19,18,24)(14,23,16,22)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(9,14)(10,11)(12,18)(13,22)(15,16)(17,24)(19,20)(21,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=g,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, 7881152);