#7901874

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

4370 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)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,29)(28,36)(30,32)(31,34)(33,38)(35,37)(39,40);(1,2)(3,7)(4,8)(5,6)(25,27)(26,31)(28,35)(29,32)(30,34)(33,37)(36,39)(38,40);(10,14)(15,17)(16,21)(22,23)(26,32)(27,34)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,31)(28,35)(29,32)(30,34)(33,37)(36,39)(38,40);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,38)(26,33,32,36)(27,35,34,40)(29,39,31,37);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,31)(26,27,32,34)(28,37,38,39)(33,40,36,35)

Polycyclic presentation:

2^9|d^2=gi,e^2=g,f^2=h,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, 7809070);