#9863065

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 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)(15,16)(17,18,23,20)(19,21,24,26)(22,25,31,28)(27,29,32,30)(35,37)(39,42)(43,44)(46,47);(1,2,6,3)(4,9,12,7)(5,8,14,10)(11,16,13,15)(19,24)(21,26)(27,32)(29,30)(33,34)(35,42)(36,40)(37,39)(38,41)(43,47)(44,46)(45,48);(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37,38,35)(34,39,41,42)(36,44,45,43)(40,46,48,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9770261);