#8429462

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,11)(4,8)(5,6)(7,10)(9,16)(12,15)(13,14)(18,23)(19,20)(26,29)(27,28)(34,40)(36,41)(42,48)(44,46);(3,4)(8,11)(12,14)(13,15)(17,18,22,23)(19,24,20,25)(21,26,30,29)(27,31,28,32)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,46)(47,48);(1,3,7,4)(2,8,10,11)(5,12,16,14)(6,13,9,15)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,35,39,37)(34,36,40,41)(38,43,47,45)(42,44,48,46);(1,4,7,3)(2,8,10,11)(5,14,16,12)(6,13,9,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,36,39,41)(34,35,40,37)(38,46,47,44)(42,45,48,43);(1,5)(2,9)(3,12)(4,14)(6,10)(7,16)(8,15)(11,13)(17,20,22,19)(18,25,23,24)(21,28,30,27)(26,32,29,31)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,46,48,44);(1,6,7,9)(2,5,10,16)(3,13,4,15)(8,12,11,14)(17,21,22,30)(18,26,23,29)(19,27,20,28)(24,31,25,32)(33,38,39,47)(34,42,40,48)(35,43,37,45)(36,44,41,46)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8336658);