#136917

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, prime radical	C₂³	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 386 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) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=h,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, 44113);