#4714008

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1308 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4621204);