#8943849

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	#40640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851045);