#9685541

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	#54620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=g,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, 9592737);