#136985

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gi,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44181);