#99705

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#386
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 380 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=g,e^2=hi,f^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh

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, 6901);