#7848820

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

4374 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=gh,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, 7756016);