#108623

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

490 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 158 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 — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=di,d^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,e^c=eh,f^c=fi

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, 15819);