#7919663

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

4591 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)(11,19)(12,21)(13,18)(14,16)(15,20)(17,22)(23,24)(25,26,31,30)(27,34,29,32)(28,33,38,37)(35,40,36,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24);(10,16)(11,20)(17,21)(18,24)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,12,14,22)(10,17,16,21)(11,18,20,24)(13,19,23,15)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,17)(12,15)(14,23)(16,24)(19,22)(20,21)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=gh,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, 7826859);