#8843579

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

4044 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(20,27)(23,25)(28,32)(30,31)(33,34,39,44)(35,42,37,40)(36,41,47,45)(38,46,48,43);(2,6)(7,9)(8,12)(15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(35,37)(38,48)(40,42)(43,46);(3,10)(7,9)(11,13)(15,16)(18,19)(23,25)(24,26)(30,31)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)

Polycyclic presentation:

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

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, 8750775);