#8975233

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	#44328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=gh,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, 8882429);