#9206457

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113653);