#8125462

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	#53129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032658);