#7971652

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	#45737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7878848);