#8534887

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	#52718	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8442083);