#507576

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265562, #72265596, #72265609, …
maximal normal subgroup of	#72265562, #72265596, #72265609, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414772);