#7986020

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	#49574	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4416 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=g,f^2=gi,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, 7893216);