#8919498

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	#54963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gh,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, 8826694);