#7973969

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	#49575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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)(9,10,15,17)(11,20,23,16)(12,18,24,14)(13,21,22,19)(26,30)(27,33)(32,36)(35,37);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,21)(13,18)(14,22)(15,23)(17,20)(19,24)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(10,17)(12,24)(16,20)(19,21)(25,26,29,30)(27,34,33,31)(28,32,38,36)(35,40,37,39);(9,12)(10,18)(11,21)(13,16)(14,17)(15,24)(19,23)(20,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,13,15,22)(10,19,17,21)(11,18,23,14)(12,16,24,20)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,12,17,24)(11,22,23,13)(16,21,20,19)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,e^2=g,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, 7881165);