#491468

order	512
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=fh,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225177
maximal normal subgroup of	#72225177
maximal quotient of	#72225200, #72225204, #72225207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398664);