#101694

order	512
nilpotency class	3
isoclinism	isoclinic class of #101687
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

459 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gh,e^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8890);