#8549565

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,e^2=h,f^2=ghi,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, 8456761);