#8818108

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	#50100	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725304);