#7954730

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	#59026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4575 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)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,24,20,22)(28,36)(32,37)(34,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,24,21,22)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(10,16)(11,12)(18,21)(19,20)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,40)(37,39);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,22,21,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4,5,2)(3,7,8,6)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,24,17,22)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,40)(37,38);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,20,21,19)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=g,f^2=i,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, 7861926);