#137252

order	512
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

375 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,10,14,16)(4,13,15,7)(5,8)(6,9)(11,12)(17,18,22,28,23,29,38,45,24,30,36,43,20,26,21,27)(19,31,34,42,32,44,33,46,35,47,37,25,39,41,48,40);(1,3,11,13,6,14,5,7)(2,4,12,10,9,15,8,16)(17,19,20,32,24,35,23,39)(18,25,26,40,30,42,29,46)(21,37,36,48,38,34,22,33)(27,44,43,47,45,41,28,31);(1,4)(2,7)(3,12)(5,16)(6,15)(8,14)(9,13)(10,11)(17,20,24,23)(18,26,30,29)(19,32,35,39)(21,36,38,22)(25,40,42,46)(27,43,45,28)(31,44,47,41)(33,37,48,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,f^2=gh,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,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, 44448);