#136612

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

462 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 126 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)(6,9,11,12)(13,14,18,24,19,25,35,23,20,26,33,42,16,22,17,34)(15,27,30,21,28,38,29,37,32,43,44,40,31,41,36,39);(5,6,10,9,8,11,7,12)(13,15,16,28,20,32,19,31)(14,21,25,39,26,40,22,37)(17,30,33,29,35,44,18,36)(23,41,42,43,34,38,24,27);(2,4)(13,16,20,19)(14,22,26,25)(15,28,32,31)(17,33,35,18)(21,37,40,39)(23,24,34,42)(27,38,43,41)(29,44,36,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gi,h^b=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, 43808);