#7974480

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	#44234	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4479 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=g,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, 7881676);