#7893678

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 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)(25,26,31,30)(27,34,29,32)(28,33,38,37)(35,40,36,39);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,18,19)(12,20,23,13)(14,21,24,17)(25,27,31,29)(26,32,30,34)(28,35,38,36)(33,39,37,40);(1,2,5,4)(3,6,8,7)(9,12,15,23)(10,17,18,21)(11,13,22,20)(14,16,24,19)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,12)(10,17)(11,20)(13,22)(14,16)(15,23)(18,21)(19,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,13,15,20)(10,14,18,24)(11,12,22,23)(16,21,19,17)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(1,4,5,2)(3,7,8,6)(9,14)(10,13)(11,21)(12,16)(15,24)(17,22)(18,20)(19,23)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=hi,e^2=h,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, 7800874);