#293788

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7)(4,5)(6,8)(9,10,13,17)(11,18,21,15)(12,22,14,16)(19,24,20,23)(25,26)(27,32)(28,37)(29,39)(30,34)(31,35)(33,36)(38,40);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,18)(22,24)(25,27)(26,32)(28,29)(30,38)(31,36)(33,35)(34,40)(37,39);(11,19)(12,14)(15,23)(16,22)(18,24)(20,21)(25,28,30,27,31,38,29,36)(26,33,39,40,35,32,34,37);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,36)(28,38)(32,33)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 200984);