#122812

order	512
nilpotency class	4
isoclinism	isoclinic class of #122724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

376 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 68 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)(9,10,14,20,16,22,15,21)(11,23,26,33,28,38,27,37)(12,29,13,32,31,19,30,18)(17,24,34,25,36,40,35,39);(2,5)(3,8)(6,7)(9,11,16,28)(10,17,22,36)(12,25,31,39)(13,24,30,40)(14,27,15,26)(18,23,32,38)(19,33,29,37)(20,35,21,34);(2,6)(3,8)(9,12,15,30,16,31,14,13)(10,18,20,29,22,32,21,19)(11,24,27,39,28,40,26,25)(17,33,34,38,36,37,35,23)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ghi,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cfhi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30008);