#8119270

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	#55700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026466);