#99422

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle	C₂³	#385
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

345 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=gh,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dg,e^b=eh,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6618);