#7720176

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

4549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7627372);