#475838

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi

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, 383034);