#7893271

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

4380 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)(9,10,15,17)(11,18,14,16)(12,13,22,24)(19,20,21,23)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,17,18)(12,21,22,19)(13,20,24,23);(1,2,5,4)(3,6,8,7)(9,12)(10,13)(11,19)(14,21)(15,22)(16,23)(17,24)(18,20)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(9,13)(10,12)(11,20)(14,23)(15,24)(16,21)(17,22)(18,19)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(9,12)(10,13)(11,21)(14,19)(15,22)(16,20)(17,24)(18,23)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,16,17,18)(12,21,22,19)(13,23,24,20)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7800467);