#138122

order	512
nilpotency class	4
isoclinism	isoclinic class of #137980
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	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=ghi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

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, 45318);