#8898525

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	#46521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4155 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)(3,7)(4,13)(5,8)(6,10)(9,12)(11,16)(14,15)(17,18,24,28)(19,29,32,25)(20,26,21,23)(22,27,31,30)(33,34)(35,42)(36,45)(37,41)(38,39)(40,44)(43,47)(46,48);(1,3)(2,7)(4,14)(5,11)(6,12)(8,16)(9,10)(13,15)(17,19)(18,25)(20,27)(21,30)(22,26)(23,31)(24,32)(28,29)(33,35)(34,40)(36,43)(37,46)(38,42)(39,44)(41,47)(45,48);(1,4,6,5)(2,8,10,13)(3,11,12,14)(7,15,9,16)(17,20,24,21)(18,26,28,23)(19,27,32,30)(22,29,31,25)(33,36,39,37)(34,41,38,45)(35,43,44,46)(40,47,42,48);(1,5,6,4)(2,8,10,13)(3,11,12,14)(7,16,9,15)(17,21,24,20)(18,26,28,23)(19,30,32,27)(22,29,31,25)(33,35)(34,40)(36,43)(37,46)(38,42)(39,44)(41,47)(45,48);(1,2)(3,9)(4,8)(5,13)(6,10)(7,12)(11,16)(14,15)(17,22,24,31)(18,27,28,30)(19,26,32,23)(20,29,21,25)(33,37,39,36)(34,41,38,45)(35,43,44,46)(40,48,42,47);(1,3)(2,9)(4,14)(5,11)(6,12)(7,10)(8,15)(13,16)(17,23,24,26)(18,20,28,21)(19,31,32,22)(25,27,29,30)(33,38,39,34)(35,42,44,40)(36,41,37,45)(43,47,46,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=g,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, 8805721);