#9863142

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

4085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,6)(3,9,4,8)(5,10,13,14)(11,16,12,15)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34,39,42)(35,40,38,48)(36,41,46,37)(43,44,45,47);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,36,39,46)(34,37,42,41)(35,43,38,45)(40,47,48,44);(1,4,7,3)(2,9,6,8)(5,12,13,11)(10,16,14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770338);