#8994651

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	#52310	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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:

(1,2,7,5)(3,4,11,8)(6,13,14,9)(10,15,16,12)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32)(33,34)(35,40)(36,42)(37,38)(39,44)(41,45)(43,48)(46,47);(1,3,7,11)(2,8,5,4)(6,10,14,16)(9,15,13,12)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(33,35,39,46)(34,40,44,47)(36,43,45,37)(38,42,48,41);(2,5)(4,8)(6,14)(10,16)(20,31)(21,30)(22,28)(25,32)(33,36,39,45)(34,41,44,42)(35,37,46,43)(38,40,48,47);(1,4)(2,3)(5,11)(6,12)(7,8)(9,10)(13,16)(14,15)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,36)(34,42)(35,43)(37,46)(38,47)(39,45)(40,48)(41,44);(1,5,7,2)(3,8,11,4)(6,13,14,9)(10,15,16,12)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,37,39,43)(34,38,44,48)(35,45,46,36)(40,41,47,42);(1,6,7,14)(2,9,5,13)(3,10,11,16)(4,12,8,15)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(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=gi,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, 8901847);