#7918669

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

4350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=ghi,e^2=i,f^2=gi,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, 7825865);