#503514

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

Aliases

No aliases for this group.

Subgroups

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

830 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=ef,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^c=di

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, 410710);