#7848822

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

4355 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)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(25,26)(27,29)(28,33)(30,34)(31,35)(32,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,24)(21,22)(25,27,31,37)(26,32,35,29)(28,38,30,36)(33,39,34,40);(10,12)(11,14)(17,20)(19,21)(25,28,31,30)(26,33,35,34)(27,36,37,38)(29,39,32,40);(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,29)(26,27)(28,39)(30,40)(31,32)(33,36)(34,38)(35,37);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,30,31,28)(26,34,35,33)(27,36,37,38)(29,39,32,40);(1,3,4,7)(2,5,6,8)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,27)(26,29)(28,38)(30,36)(31,37)(32,35)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=h,e^2=i,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, 7756018);