#7844580

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

4627 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)(9,10,14,15)(11,16,17,12)(13,18,22,23)(19,24,21,20)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,21)(14,17)(15,16)(18,24)(19,22)(20,23)(25,27,31,37)(26,32,35,30)(28,38,29,36)(33,39,34,40);(10,15)(11,17)(13,22)(20,24)(25,28,31,29)(26,33,35,34)(27,36,37,38)(30,39,32,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,24)(20,23)(21,22)(25,29,31,28)(26,34,35,33)(27,36,37,38)(30,39,32,40);(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,30)(26,27)(28,39)(29,40)(31,32)(33,36)(34,38)(35,37);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,27)(26,30)(28,38)(29,36)(31,37)(32,35)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,e^2=h,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, 7751776);