#416566

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1678 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1678 conjugacy classes of subgroups — too many to draw. Only its 274 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323762);