#135001

order	512
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂²×D₄	Q₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^c=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42197);