#10247280

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

4143 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)(3,10,5,8)(4,12,14,9)(11,15,13,16)(18,22)(20,28)(23,25)(26,29)(33,34,39,37)(35,40,45,46)(36,42,47,41)(38,44,48,43);(1,3,7,5)(2,8,6,10)(4,13,14,11)(9,16,12,15)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31)(33,35,39,45)(34,40,37,46)(36,43,47,44)(38,41,48,42);(17,18,21,22)(19,23,27,25)(20,24,28,30)(26,31,29,32)(33,36,39,47)(34,41,37,42)(35,44,45,43)(38,46,48,40);(1,4,7,14)(2,9,6,12)(3,11,5,13)(8,15,10,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,36)(34,42)(35,44)(37,41)(38,46)(39,47)(40,48)(43,45);(1,5,7,3)(2,8,6,10)(4,11,14,13)(9,16,12,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,37,39,34)(35,40,45,46)(36,42,47,41)(38,43,48,44);(1,6,7,2)(3,10,5,8)(4,12,14,9)(11,16,13,15)(17,20,21,28)(18,24,22,30)(19,26,27,29)(23,31,25,32)(33,38)(34,43)(35,41)(36,46)(37,44)(39,48)(40,47)(42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154476);