#8159286

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,4)(8,9)(10,12)(15,16)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(34,35)(39,41)(40,42)(46,47);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,7,2)(3,8,5,9)(6,12,14,10)(11,15,13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

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

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, 8066482);