#8230659

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	#58071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

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

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, 8137855);