#7973976

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	#43038	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4532 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)(11,19)(12,15)(13,22)(14,18)(16,20)(17,24)(21,23)(25,26,31,30)(27,35,36,32)(28,38,39,33)(29,34,40,37);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,19)(13,21)(14,24)(15,20)(17,23)(18,22)(25,27,31,36)(26,32,30,35)(28,37,39,34)(29,38,40,33);(1,3,5,4)(2,7,6,8)(10,12)(11,20)(13,24)(18,23)(27,36)(29,40)(32,35)(34,37);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,19,20,16)(13,22,24,17)(14,23,21,18)(25,28,31,39)(26,33,30,38)(27,37,36,34)(29,35,40,32);(1,3,5,4)(2,7,6,8)(9,13,15,24)(10,17,12,22)(11,21,20,14)(16,23,19,18)(25,29,31,40)(26,34,30,37)(27,38,36,33)(28,35,39,32);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,18,12,23)(11,13,20,24)(16,17,19,22)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,34,40,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,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, 7881172);