#7891196

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	#35393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4461 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)(9,10,15,12)(11,18,16,13)(14,17,23,20)(19,22,24,21)(25,26,31,33)(27,35,38,32)(28,29,39,40)(30,34,37,36);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,27,31,38)(26,32,33,35)(28,36,39,34)(29,37,40,30);(1,2,5,4)(3,6,8,7)(10,12)(11,16)(17,20)(19,24)(26,33)(28,39)(32,35)(34,36);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21)(25,28)(26,29)(27,36)(30,35)(31,39)(32,37)(33,40)(34,38);(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,29)(26,28)(27,37)(30,38)(31,40)(32,36)(33,39)(34,35);(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(18,22)(25,30,31,37)(26,34,33,36)(27,29,38,40)(28,35,39,32)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7798392);