#141067

order	512
nilpotency class	4
isoclinism	isoclinic class of #125269
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4)(5,6,10,15,9,14,27,30,11,17,29,34,24,16,35,33)(7,18,21,36,20,25,32,28,22,12,13,31,19,8,23,26);(5,7,10,23,9,19,27,13,11,22,29,32,24,20,35,21)(6,12,30,36,16,8,15,28,17,18,33,31,14,25,34,26);(1,2,3,4)(5,8,24,12,11,25,9,18)(6,13,14,23,17,21,16,32)(7,15,20,30,22,34,19,33)(10,28,35,36,29,26,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48263);