#134641

order	512
nilpotency class	4
isoclinism	isoclinic class of #134475
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=i,e^2=hi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308779, #72308791, #72308804
maximal normal subgroup of	#72308779, #72308791, #72308804

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41837);