#8806775

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

4231 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)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,18)(19,21)(20,24)(22,26)(23,28)(25,30)(27,31)(29,32)(33,34,40,44)(35,37,38,41)(36,43,39,42)(45,46,48,47);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,19,22,28)(18,23,26,21)(20,27,30,32)(24,29,25,31)(33,35)(34,41)(36,45)(37,44)(38,40)(39,48)(42,46)(43,47);(1,2,5,8)(3,9,11,6)(4,7,14,12)(10,16,13,15)(17,20,22,30)(18,24,26,25)(19,27,28,32)(21,31,23,29)(33,36,40,39)(34,42,44,43)(35,45,38,48)(37,47,41,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,26)(28,32)(33,37,40,41)(34,35,44,38)(36,46,39,47)(42,48,43,45);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,21)(18,19)(20,29)(22,23)(24,32)(25,27)(26,28)(30,31)(33,38,40,35)(34,41,44,37)(36,45,39,48)(42,47,43,46);(1,4,5,14)(2,7,8,12)(3,10,11,13)(6,15,9,16)(17,19)(18,21)(20,27)(22,28)(23,26)(24,31)(25,29)(30,32)(33,39,40,36)(34,43,44,42)(35,45,38,48)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=hi,e^2=hi,f^2=h,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, 8713971);