#136983

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

421 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 112 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)(13,14,18,23,33,42,31,40,19,24,17,22,16,21,32,41)(15,25,28,38,26,44,34,20,30,37,27,36,29,43,35,39);(1,2,3,4)(6,10)(7,8)(11,12)(13,15)(14,20)(16,26)(17,34)(18,35)(19,30)(21,36)(22,43)(23,44)(24,39)(25,40)(27,31)(28,32)(29,33)(37,41)(38,42);(6,10)(7,8)(11,12)(13,16,19,33)(14,21,24,42)(15,26,30,29)(17,31,18,32)(20,36,39,38)(22,40,23,41)(25,44,37,43)(27,35,28,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44179);