#7893793

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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)(9,10,15,18)(11,19,22,16)(12,21,23,17)(13,24,20,14)(25,26,32,29)(27,36,38,33)(28,39,31,34)(30,40,37,35);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,18,19)(12,20,23,13)(14,21,24,17)(25,27,32,38)(26,33,29,36)(28,37,31,30)(34,35,39,40);(9,12,15,23)(10,17,18,21)(11,13,22,20)(14,16,24,19)(25,28,32,31)(26,34,29,39)(27,37,38,30)(33,35,36,40);(9,12)(10,17)(11,20)(13,22)(14,16)(15,23)(18,21)(19,24)(25,29,32,26)(27,36,38,33)(28,34,31,39)(30,40,37,35);(9,13,15,20)(10,14,18,24)(11,12,22,23)(16,21,19,17)(25,30)(26,35)(27,28)(29,40)(31,38)(32,37)(33,39)(34,36);(1,4,5,2)(3,7,8,6)(9,14)(10,13)(11,21)(12,16)(15,24)(17,22)(18,20)(19,23)(25,31,32,28)(26,34,29,39)(27,30,38,37)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=i,f^2=gi,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, 7800989);