#304231

order	512
nilpotency class	3
isoclinism	isoclinic class of #267755
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

806 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dh,g^c=gi

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, 211427);