#8873641

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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)(3,6)(4,12)(5,9)(7,11)(8,14)(10,16)(13,15)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(34,38)(35,41)(40,43)(42,47);(1,3,5,11)(2,6,9,7)(4,13,14,10)(8,16,12,15)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,16)(12,15)(13,14)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,16)(12,15)(13,14)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48);(1,4,5,14)(2,8,9,12)(3,10,11,13)(6,15,7,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780837);