#8919298

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826494);