#508930

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 812 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)(7,10)(8,9)(11,12)(13,14,19,30,29,39,28,21,17,23,16,22,18,24,32,38)(15,20,31,43,40,44,42,37,25,35,26,34,27,36,41,33);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,25)(18,27)(19,31)(21,37)(22,34)(23,35)(24,36)(28,42)(29,40)(30,43)(32,41)(33,38)(39,44);(2,4)(7,11)(10,12)(14,21)(15,25)(16,28)(18,29)(19,32)(20,33)(22,24)(23,38)(26,41)(30,39)(31,42)(34,44)(35,37)(36,43);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,h^c=hi

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, 416126);