#138127

order	512
nilpotency class	4
isoclinism	isoclinic class of #125076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

404 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 118 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)(7,11)(10,12)(13,14,18,24,19,25,34,41,20,26,32,39,16,22,17,23)(15,27,30,38,28,40,29,42,31,43,33,21,35,37,44,36);(5,6,7,9,8,10,11,12)(13,15,19,35,20,31,16,28)(14,21,25,42,26,38,22,36)(17,33,18,29,34,30,32,44)(23,40,24,27,41,37,39,43);(2,4)(13,16,20,19)(14,22,26,25)(15,28,31,35)(17,32,34,18)(21,36,38,42)(23,39,41,24)(27,40,43,37)(29,33,44,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45323);