#6122163

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fhi,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6029359);