#238059

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

1620 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1620 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)(11,16)(12,25,28,36)(13,19)(14,18)(15,20)(17,29,34,27)(21,33,38,40)(22,35)(23,31)(24,32)(26,30,37,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,26,28,37)(13,23)(14,22)(15,24)(17,33,34,40)(18,35)(19,31)(20,32)(21,27,38,29)(25,39,36,30);(5,8)(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,30)(18,25)(19,36)(20,34)(22,37)(23,26)(24,38)(31,40)(32,39)(33,35);(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,28,29)(16,31,32,35)(17,25,34,36)(21,37,38,26)(30,40,39,33)

Polycyclic presentation:

2^9|a^2=e,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, 145255);