#9685500

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	#48518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,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, 9592696);