#112834

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1164 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1164 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh,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, 20030);