#138206

order	512
nilpotency class	4
isoclinism	isoclinic class of #125508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=i,d^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,f^b=fh,h^b=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, 45402);