#9712160

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	#53628	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=ghi,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, 9619356);