#7918495

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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=h,e^2=i,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, 7825691);