#8975239

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

4272 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(2,7)(4,14)(8,11)(10,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(2,7)(3,12)(4,14)(5,16)(19,26)(20,24)(21,29)(23,30)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8882435);