#120324

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
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

430 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 130 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)(17,18,22,26,23,28,21,27)(19,29,32,40,33,43,31,24)(20,34,38,44,37,25,36,42)(30,41,47,39,46,45,35,48);(1,3,4,11)(2,6,8,10)(5,14,13,12)(7,16,9,15)(17,19,21,31,23,33,22,32)(18,24,26,29,28,40,27,43)(20,35,36,46,37,47,38,30)(25,41,42,39,34,45,44,48);(2,7)(5,13)(6,15)(8,9)(10,16)(12,14)(17,20,21,36,23,37,22,38)(18,25,26,42,28,34,27,44)(19,30,31,35,33,46,32,47)(24,39,29,45,40,48,43,41)

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=fi,c^b=cf,e^b=ei,f^b=fh,e^c=ehi,f^c=fh

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, 27520);