#8691790

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	#58166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4142 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8598986);