#415263

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

986 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 258 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=fi,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=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, 322459);