#8550991

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458187);