#9706279

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	#55579	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,e^2=ghi,f^2=gh,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, 9613475);