#8583468

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	#41445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8490664);