#9063304

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	#51048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(3,7,10,9)(5,12,13,8)(11,15,16,14)(17,18,23,26)(19,24,27,20)(21,22,25,32)(28,31,30,29)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,44,46,47);(1,2,6,4)(3,9,10,7)(5,8,13,12)(11,15,16,14)(17,19)(18,24)(20,26)(21,30)(22,29)(23,27)(25,28)(31,32)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,44,48,47);(2,4)(3,10)(5,13)(14,15)(19,27)(20,24)(21,25)(22,32)(33,35,38,45)(34,40,42,39)(36,46,41,43)(37,47,48,44);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,20)(18,19)(21,31)(22,30)(23,24)(25,29)(26,27)(28,32)(33,35)(34,39)(36,46)(37,44)(38,45)(40,42)(41,43)(47,48);(1,4,6,2)(3,7,10,9)(5,12,13,8)(11,14,16,15)(17,21,23,25)(18,22,26,32)(19,28,27,30)(20,29,24,31)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,47,40,44);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,22)(18,25)(19,29)(20,30)(21,26)(23,32)(24,28)(27,31)(33,37)(34,41)(35,44)(36,42)(38,48)(39,43)(40,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8970500);