#10143334

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	#43373	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 10050530);