#9029107

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	#43822	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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)(3,8,6,4)(5,9,15,16)(11,13,14,12)(17,18,23,27)(19,22,21,25)(20,26,31,24)(28,30,29,32)(33,34,38,42)(35,40,45,39)(36,46,41,37)(43,44,47,48);(1,3,7,6)(2,8,10,4)(5,11,15,14)(9,13,16,12)(17,19,23,21)(18,22,27,25)(20,29,31,28)(24,30,26,32)(33,35,38,45)(34,39,42,40)(36,43,41,47)(37,44,46,48);(1,4,7,8)(2,3,10,6)(5,13,15,12)(9,14,16,11)(17,20,23,31)(18,24,27,26)(19,28,21,29)(22,32,25,30)(33,35,38,45)(34,39,42,40)(36,47,41,43)(37,48,46,44);(1,4)(2,3)(5,13)(6,10)(7,8)(9,14)(11,16)(12,15)(17,21,23,19)(18,25,27,22)(20,28,31,29)(24,32,26,30)(33,36,38,41)(34,37,42,46)(35,43,45,47)(39,44,40,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,20)(18,26)(19,28)(21,29)(22,30)(23,31)(24,27)(25,32)(33,35)(34,40)(36,47)(37,44)(38,45)(39,42)(41,43)(46,48);(1,6,7,3)(2,4,10,8)(5,14,15,11)(9,12,16,13)(17,22,23,25)(18,19,27,21)(20,30,31,32)(24,29,26,28)(33,37)(34,41)(35,44)(36,42)(38,46)(39,47)(40,43)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8936303);