#9685596

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	#43971	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9592792);