#8818106

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	#43275	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725302);