#8898519

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	#46384	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4276 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30)(33,34)(35,39)(36,45)(37,40)(38,42)(41,44)(43,48)(46,47);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,35)(34,39)(36,46)(37,43)(38,44)(40,48)(41,42)(45,47);(2,6)(8,9)(10,13)(15,16)(21,29)(24,31)(27,28)(30,32)(33,36,38,37)(34,40,42,45)(35,43,44,46)(39,47,41,48);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,37,38,36)(34,40,42,45)(35,43,44,46)(39,48,41,47);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,34)(35,41)(36,40)(37,45)(38,42)(39,44)(43,48)(46,47);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)(33,35)(34,41)(36,46)(37,43)(38,44)(39,42)(40,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=h,e^2=g,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, 8805715);