#8810189

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	#58892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=h,e^2=h,f^2=hi,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, 8717385);