#8581774

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	#56951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,23,20)(19,21,24,26)(22,25,30,28)(27,29,31,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(3,4)(7,10)(11,12)(15,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,18)(19,26,24,21)(22,25,30,28)(27,29,31,32)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gi,e^2=ghi,f^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, 8488970);