#508350

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415546);