#8898512

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	#47177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805708);