#10217050

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3919 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10124246);