#114929

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1156 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1156 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) = 256

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

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

Polycyclic presentation:

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

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, 22125);