#9545953

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	#41182	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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)(3,8,5,6)(4,9,13,16)(11,15,14,12)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(19,25)(22,29)(26,31)(30,32)(34,39)(35,42)(41,44)(43,48);(3,5)(6,8)(11,14)(12,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9453149);