#244056

order	512
nilpotency class	3
isoclinism	isoclinic class of #233501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

968 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,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, 151252);