#7954714

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

4527 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)(9,10,15,17)(11,18,12,16)(13,22,23,19)(14,20,24,21)(28,36)(32,39)(35,40)(37,38);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,18)(13,20,23,21)(14,19,24,22)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(10,17)(11,12)(19,22)(20,21)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,22,24,19)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,4,5,2)(3,7,8,6)(9,13,15,23)(10,19,17,22)(11,20,12,21)(14,16,24,18)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(9,14)(10,20)(11,19)(12,22)(13,18)(15,24)(16,23)(17,21)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,39)(36,38)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=g,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, 7861910);