#490613

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

414 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 150 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)(18,23)(20,24)(21,31)(25,33)(26,41)(27,36)(29,37)(30,43)(32,38)(34,35)(39,44)(40,48)(42,47)(45,46);(17,18,24,35,21,26,32,36,22,27,38,41,31,34,20,23)(19,25,37,46,30,40,42,44,28,39,47,48,43,45,29,33);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,25)(20,29)(21,30)(22,28)(23,33)(24,37)(26,40)(27,39)(31,43)(32,42)(34,45)(35,46)(36,44)(38,47)(41,48);(19,28)(25,39)(29,42)(30,43)(33,44)(37,47)(40,45)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=h,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, 397809);