#8772913

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	#49802	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=gh,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, 8680109);