#7736705

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

4500 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)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,33)(27,31,34,28)(29,32,38,40)(35,39,37,36);(1,3,5,8)(2,6,4,7)(25,27)(26,31)(28,33)(29,37)(30,34)(32,36)(35,38)(39,40);(1,4,5,2)(3,7,8,6)(10,14)(15,17)(16,21)(22,23)(27,34)(28,31)(35,37)(36,39);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,31)(26,27,33,34)(29,36,38,39)(32,35,40,37);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27)(26,31)(28,33)(29,35)(30,34)(32,39)(36,40)(37,38);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643901);