#7920407

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

4680 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)(9,10,14,16)(11,15,19,21)(12,20,22,17)(13,18,23,24)(25,26)(27,35)(28,38)(29,33)(30,34)(31,37)(32,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,19)(10,15,16,21)(12,18,22,24)(13,20,23,17)(25,27,30,37)(26,31,34,35)(28,36,39,29)(32,40,38,33);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(17,20)(18,24)(25,26,30,34)(27,31,37,35)(28,32,39,38)(29,33,36,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39);(9,12)(10,17)(11,18)(13,15)(14,22)(16,20)(19,24)(21,23)(25,28)(26,32)(27,36)(29,37)(30,39)(31,40)(33,35)(34,38);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,20)(12,21)(14,23)(15,22)(16,24)(17,19)(25,29,30,36)(26,33,34,40)(27,28,37,39)(31,32,35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,f^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, 7827603);