#129378

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2274 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

863 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 863 conjugacy classes of subgroups — too many to draw. Only its 122 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)(11,19,22,32)(12,23)(13,26,28,17)(14,18)(15,21)(16,25)(20,34,36,33)(24,31,40,38)(27,30)(29,39)(35,37);(2,5)(3,7)(9,11,14,22)(10,15,18,30)(12,20,25,36)(13,27,28,21)(16,29,23,37)(17,32,26,19)(24,39,40,35)(31,34,38,33);(9,12)(10,16,18,23)(11,20)(13,24)(14,25)(15,29,30,37)(17,31,26,38)(19,33,32,34)(21,35)(22,36)(27,39)(28,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fgi,e^c=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36574);