#4278385

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

1364 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)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(34,35)(36,43)(37,45)(47,48);(10,13)(14,16)(18,22)(23,30)(24,27)(31,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,48,46,47);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,48,46,47);(1,2)(3,5)(4,6)(7,8)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,36)(34,39)(35,41)(37,46)(38,43)(40,48)(42,47)(44,45);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,28)(30,32)(33,37,38,45)(34,40,35,42)(36,44,43,46)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=fg,e^2=fghi,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, 4185581);