#7973915

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	#36138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4419 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^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, 7881111);