#7848824

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

4538 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)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,24)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,13)(14,20,22,19)(18,21,24,23)(25,27)(26,32)(28,38)(29,36)(30,35)(31,37)(33,40)(34,39);(10,17)(11,12)(14,22)(21,23)(25,28,31,29)(26,33,35,34)(27,36,37,38)(30,40,32,39);(9,12,15,11)(10,16,17,13)(14,20,22,19)(18,23,24,21)(25,27)(26,30)(28,36)(29,38)(31,37)(32,35)(33,40)(34,39);(9,13,15,16)(10,12,17,11)(14,21,22,23)(18,20,24,19)(25,29,31,28)(26,34,35,33)(27,36,37,38)(30,40,32,39);(1,4,5,2)(3,7,8,6)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30)(26,27)(28,39)(29,40)(31,32)(33,38)(34,36)(35,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 7756020);