#10143447

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	#55587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,24,22)(19,27,23,25)(20,26,21,30)(28,32,29,31)(33,34,40,44)(35,41,46,48)(36,42,37,39)(38,43,45,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,24,23)(18,25,22,27)(20,28,21,29)(26,31,30,32)(33,35,40,46)(34,41,44,48)(36,47,37,43)(38,39,45,42);(2,4)(3,10)(9,12)(11,16)(17,20,24,21)(18,26,22,30)(19,28,23,29)(25,31,27,32)(33,36,40,37)(34,42,44,39)(35,43,46,47)(38,41,45,48);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,21,24,20)(18,26,22,30)(19,29,23,28)(25,31,27,32)(33,37,40,36)(34,39,44,42)(35,43,46,47)(38,41,45,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,22,24,18)(19,27,23,25)(20,30,21,26)(28,32,29,31)(33,38)(34,43)(35,39)(36,41)(37,48)(40,45)(42,46)(44,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,23,24,19)(18,25,22,27)(20,29,21,28)(26,31,30,32)(33,39)(34,36)(35,45)(37,44)(38,46)(40,42)(41,47)(43,48)

Polycyclic presentation:

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

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, 10050643);