#153002

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator, prime radical	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,f^b=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60198);