#8818344

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

4072 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(33,34,40,44)(35,42,37,38)(36,41,39,43)(45,46,47,48);(2,6)(4,13)(7,9)(11,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,35,40,37)(34,38,44,42)(36,45,39,47)(41,48,43,46);(2,6)(3,10)(8,12)(11,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,36,40,39)(34,41,44,43)(35,45,37,47)(38,48,42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,37,40,35)(34,42,44,38)(36,45,39,47)(41,48,43,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,38,40,42)(34,35,44,37)(36,46,39,48)(41,47,43,45);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,39,40,36)(34,43,44,41)(35,45,37,47)(38,48,42,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8725540);