#10195509

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	#53544	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=h,e^2=g,f^2=ghi,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, 10102705);