#491583

order	512
nilpotency class	4
isoclinism	isoclinic class of #491556
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33733	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,20,23)(19,27,29,41)(21,34,32,47)(22,35,33,26)(24,38,36,30)(25,40,37,28)(31,46,45,43)(39,44,42,48);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,23)(24,36)(25,37)(26,35)(27,41)(34,47)(39,42)(43,46);(17,19,28,44,22,31,32,38,20,29,40,48,33,45,21,30)(18,24,34,43,26,39,37,41,23,36,47,46,35,42,25,27);(3,10)(6,15)(9,16)(11,14)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398779);