#7918702

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

4589 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)(11,13)(12,20)(14,16)(15,18)(17,22)(19,21)(23,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,16)(14,18)(17,21)(20,23)(22,24);(1,3,5,4)(2,7,6,8)(10,16)(11,18)(17,20)(19,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(9,11)(10,15)(12,19)(13,16)(14,18)(17,21)(20,23)(22,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,17,16,20)(11,19,18,24)(13,23,15,21)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13,14,15)(10,11,16,18)(12,21,22,23)(17,24,20,19)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,e^2=g,f^2=ghi,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, 7825898);