#119289

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 694 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) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26485);