#325654

order	512
nilpotency class	3
isoclinism	isoclinic class of #278051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

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

normal subgroups only

This group has 597 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232850);