#139376

order	512
nilpotency class	4
isoclinism	isoclinic class of #139358
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

311 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,8,4,9)(3,11,14,7,13,16,12,15)(17,18)(19,28)(20,33,36,44)(21,31)(22,26)(23,30)(24,27)(25,35,43,41)(29,34,42,37)(32,38)(39,40)(45,46,48,47);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,16)(10,11)(17,19,28,18,23,31,39,26,24,32,38,27,22,40,21,30)(20,29,45,25,35,47,34,44,36,42,48,43,41,46,37,33);(2,8)(4,5)(7,15)(12,14)(17,20)(18,25)(19,29)(21,37)(22,41)(23,35)(24,36)(26,44)(27,43)(28,45)(30,33)(31,47)(32,42)(34,39)(38,48)(40,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46572);