#138187

order	512
nilpotency class	4
isoclinism	isoclinic class of #125438
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45383);