#241270

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 218 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)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,30,34)(27,31,37,40)(28,33,29,32)(35,39,36,38);(1,3,5,4)(2,6,7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,37)(26,31,34,40)(28,36,29,35)(32,39,33,38);(1,4,5,3)(2,6,7,8)(10,16)(11,19)(12,13)(17,18)(20,21)(22,23)(26,32)(28,29)(31,38)(33,34)(35,36)(39,40);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28,30,29)(26,33,34,32)(27,35,37,36)(31,39,40,38)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=e,d^2=h,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, 148466);