#136281

order	512
nilpotency class	4
isoclinism	isoclinic class of #12096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂²	#1867
prime radical	C₂³	#385
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=g,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=ehi,d^c=dh,e^c=eg,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43477);