#503308

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 726 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fhi,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410504);