#251665

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59303	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

946 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158861);