#507697

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 508 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=g,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,h^c=hi

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, 414893);