#8889049

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gh,e^2=gi,f^2=hi,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, 8796245);