#9730604

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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)(3,8,13,7)(4,5,9,15)(11,14,16,12)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,16)(14,15)(17,19,23,28)(18,24,25,32)(20,29,31,22)(21,27,26,30)(35,41)(38,45)(42,47)(46,48);(18,25)(19,28)(20,31)(27,30)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,6,9)(2,5,10,15)(3,11,13,16)(7,12,8,14)(17,20)(18,26)(19,29)(21,25)(22,28)(23,31)(24,30)(27,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,8)(4,11)(5,14)(6,13)(7,10)(9,16)(12,15)(17,21,23,26)(18,20,25,31)(19,30,28,27)(22,32,29,24)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5)(2,9)(3,12)(4,10)(6,15)(7,16)(8,11)(13,14)(17,22,23,29)(18,27,25,30)(19,20,28,31)(21,24,26,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9637800);