#10143282

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,22,24)(19,20,27,23)(21,25,31,32)(26,28,29,30)(33,34)(35,44)(36,41)(37,47)(38,42)(39,43)(40,46)(45,48);(1,3,7,4)(2,8,9,10)(5,12,15,13)(6,11,16,14)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,35)(34,40)(36,38)(37,45)(39,46)(41,48)(42,47)(43,44);(2,9)(3,4)(11,14)(12,13)(18,24)(20,23)(25,32)(28,30)(33,36,39,37)(34,41,43,47)(35,38,46,45)(40,48,44,42);(1,4,7,3)(2,10,9,8)(5,13,15,12)(6,14,16,11)(17,20,22,23)(18,19,24,27)(21,30,31,28)(25,29,32,26)(33,37,39,36)(34,41,43,47)(35,45,46,38)(40,48,44,42);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,21,22,31)(18,25,24,32)(19,26,27,29)(20,28,23,30)(33,34)(35,40)(36,41)(37,47)(38,48)(39,43)(42,45)(44,46);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,38,39,45)(34,42,43,48)(35,36,46,37)(40,47,44,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gi,e^2=h,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, 10050478);