#8994649

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	#52308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=ghi,e^2=h,f^2=g,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, 8901845);