#8993165

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	#41310	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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)(3,9,10,4)(5,7,13,14)(11,15,16,12)(17,18,22,24)(19,27,28,23)(20,29,30,25)(21,26,31,32)(33,34,39,43)(35,38,37,41)(36,42,47,40)(44,46,45,48);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,28)(18,23,24,27)(20,21,30,31)(25,26,29,32)(33,35,39,37)(34,38,43,41)(36,45,47,44)(40,46,42,48);(3,10)(4,9)(5,13)(7,14)(18,24)(20,30)(21,31)(23,27)(33,36,39,47)(34,40,43,42)(35,44,37,45)(38,48,41,46);(1,3,6,10)(2,4,8,9)(5,11,13,16)(7,12,14,15)(17,20,22,30)(18,25,24,29)(19,21,28,31)(23,26,27,32)(33,37,39,35)(34,41,43,38)(36,44,47,45)(40,48,42,46);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21)(18,26)(19,20)(22,31)(23,25)(24,32)(27,29)(28,30)(33,36)(34,42)(35,44)(37,45)(38,46)(39,47)(40,43)(41,48);(1,5,6,13)(2,7,8,14)(3,11,10,16)(4,12,9,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,30)(31,32)(33,38,39,41)(34,35,43,37)(36,46,47,48)(40,45,42,44)

Polycyclic presentation:

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

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, 8900361);