#7848832

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

4543 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)(11,18)(12,15)(13,21)(14,16)(17,22)(19,23)(20,24)(25,26,31,29)(27,30,33,34)(28,32,37,38)(35,36,39,40);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,16,18)(13,19,22,20)(17,23,21,24)(25,27,31,33)(26,30,29,34)(28,35,37,39)(32,36,38,40);(10,16)(11,12)(17,21)(19,20)(25,26,31,29)(27,34,33,30)(28,32,37,38)(35,40,39,36);(9,12,14,11)(10,15,16,18)(13,19,22,20)(17,24,21,23)(25,28)(26,32)(27,35)(29,38)(30,36)(31,37)(33,39)(34,40);(1,2,5,4)(3,6,8,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,29,31,26)(27,34,33,30)(28,32,37,38)(35,36,39,40);(1,4,5,2)(3,7,8,6)(9,10)(11,18)(12,15)(13,17)(14,16)(19,24)(20,23)(21,22)(25,30,31,34)(26,33,29,27)(28,36,37,40)(32,39,38,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=h,e^2=gi,f^2=gi,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, 7756028);