#120329

order	512
nilpotency class	3
isoclinism	isoclinic class of #10530
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

444 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,10)(4,8)(5,7)(6,11)(9,13)(12,16)(14,15)(17,18,22,26,23,27,21,38)(19,28,32,40,31,41,30,24)(20,33,37,44,36,25,35,43)(29,42,45,48,46,47,34,39);(1,3,4,11)(2,6,8,10)(5,14,13,12)(7,16,9,15)(17,19,21,30,23,31,22,32)(18,24,26,28,27,40,38,41)(20,34,35,46,36,45,37,29)(25,42,43,48,33,47,44,39);(2,7)(5,13)(6,15)(8,9)(10,16)(12,14)(17,20,21,35,23,36,22,37)(18,25,26,43,27,33,38,44)(19,29,30,34,31,46,32,45)(24,39,28,42,40,48,41,47)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=di,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,f^b=fh,e^c=ehi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27525);