#7849014

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

4477 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7756210);