#4615786

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

1418 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)(9,10,12,14)(11,13,15,16)(18,22)(23,28)(24,29)(30,32)(33,34,37,38)(35,41,42,39)(36,40,45,46)(43,48,44,47);(3,7)(5,8)(9,11,12,15)(10,13,14,16)(19,25)(23,28)(26,31)(30,32)(34,38)(35,42)(40,46)(43,44);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,37,42)(34,39,38,41)(36,44,45,43)(40,48,46,47);(9,10)(11,13)(12,14)(15,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,48)(45,46);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,45)(34,40,38,46)(35,43,42,44)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=fg,e^2=g,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, 4522982);