#9206577

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113773);