#149665

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 270 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) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,c^b=ci,e^b=ef

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, 56861);