#10166291

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	#43255	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

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

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, 10073487);