#294997

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 786 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) = 256

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

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

Polycyclic presentation:

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

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, 202193);