#10168482

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	#56200	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=ghi,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, 10075678);