#116368

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1444 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1444 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23564);