#4278487

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=f,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4185683);