#8898923

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	#47010	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806119);