#8127223

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

4300 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:

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

Polycyclic presentation:

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

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, 8034419);