#403836

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(7,11)(9,12)(13,14,18,25,19,26,20,24)(15,23,17,21,28,37,29,38)(16,30,34,41,35,42,33,22)(27,39,43,44,31,36,32,40);(5,6,8,10)(7,12,11,9)(13,15,18,17,19,28,20,29)(14,21,24,23,26,38,25,37)(16,31,33,43,35,27,34,32)(22,39,30,44,41,36,42,40);(5,7,8,11)(6,9,10,12)(13,16,19,35)(14,22,26,41)(15,27,28,31)(17,32,29,43)(18,33,20,34)(21,36,38,39)(23,40,37,44)(24,30,25,42);(2,4)(13,17,18,28,19,29,20,15)(14,23,24,38,26,37,25,21)(16,32,33,31,35,43,34,27)(22,40,30,39,41,44,42,36)

Polycyclic presentation:

2^9|a^2=fgi,b^2=egi,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,f^c=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, 311032);