#490617

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^c=di,f^c=fh

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, 397813);