#7918494

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

4436 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=h,e^2=gi,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, 7825690);