#241629

order	512
nilpotency class	3
isoclinism	isoclinic class of #233789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

784 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,6)(4,5)(7,8)(9,10,13,17)(11,19,21,23)(12,16,14,18)(15,22,24,20)(25,26)(27,34)(28,33)(29,31)(30,38)(32,37)(35,39)(36,40);(2,5)(6,8)(9,10)(11,19)(12,18)(13,17)(14,16)(15,22)(20,24)(21,23)(26,31)(28,30)(34,40)(35,37);(9,11,13,21)(10,15,17,24)(12,22,14,20)(16,19,18,23)(25,27,29,36)(26,32,31,39)(28,37,30,35)(33,40,38,34);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,16,17,18)(11,20,21,22)(15,23,24,19)(25,28,29,30)(26,33,31,38)(27,35,36,37)(32,34,39,40)

Polycyclic presentation:

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

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, 148825);