#138188

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=ghi,f^2=hi,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, 45384);