#8159080

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

3839 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)(4,5)(6,14)(7,8)(10,15)(11,12)(13,16)(17,18,22,26)(19,25,21,23)(20,24,30,28)(27,32,29,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32)(34,35)(39,41)(40,42)(46,47);(2,8)(4,9)(10,14)(12,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,32)(29,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)(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=gi,c^2=i,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, 8066276);