#8975404

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=h,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, 8882600);