#9206453

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3801 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,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, 9113649);