#7724558

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	#36758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4470 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)(25,26)(27,36)(28,34)(29,39)(30,35)(31,33)(32,37)(38,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,27)(26,32)(28,30)(29,34)(31,37)(33,36)(35,40)(38,39);(1,2,5,4)(3,6,8,7)(10,14)(15,21)(16,19)(22,23)(26,33)(27,37)(28,38)(29,35);(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,31,38)(26,34,33,40)(27,30,37,39)(29,36,35,32);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,29,31,35)(26,30,33,39)(27,34,37,40)(28,36,38,32);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30)(26,35)(27,38)(28,37)(29,33)(31,39)(32,34)(36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7631754);