#8876164

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	#38628	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(7,10)(11,15)(14,16)(17,18,23,28)(19,26,21,24)(20,25,31,29)(22,30,32,27)(33,34)(35,41)(36,38)(37,44)(39,43)(40,45)(42,46)(47,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(19,21)(22,32)(24,26)(27,30)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,6)(7,10)(8,12)(14,16)(17,19,23,21)(18,24,28,26)(20,30,31,27)(22,29,32,25)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,25)(19,27)(21,30)(22,26)(23,31)(24,32)(28,29)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,19)(18,26,28,24)(20,27,31,30)(22,25,32,29)(33,35)(34,41)(36,39)(37,42)(38,43)(40,47)(44,46)(45,48);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,22,23,32)(18,27,28,30)(19,29,21,25)(20,26,31,24)(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=g,c^2=h,e^2=gh,f^2=gi,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, 8783360);