#7891220

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	#59837	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4385 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)(11,16)(12,14)(13,23)(15,19)(17,22)(18,20)(21,24)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,19,24)(12,20,22,13)(14,18,17,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,2,5,4)(3,6,8,7)(9,12,15,22)(10,17,19,14)(11,13,21,20)(16,18,24,23)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(9,12)(10,14)(11,20)(13,21)(15,22)(16,18)(17,19)(23,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,13,15,20)(10,18,19,23)(11,12,21,22)(14,16,17,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38);(1,4,5,2)(3,7,8,6)(9,14)(10,12)(11,18)(13,24)(15,17)(16,20)(19,22)(21,23)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=hi,e^2=i,f^2=ghi,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, 7798416);