#7896056

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4424 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7803252);