#238060

order	512
nilpotency class	3
isoclinism	isoclinic class of #238058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1612 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1612 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,f^c=fh,g^c=gh

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, 145256);