#506323

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321154
maximal normal subgroup of	#72321154

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413519);