#119407

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

682 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,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, 26603);