#241241

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 818 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) = 256

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

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

Polycyclic presentation:

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

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, 148437);