#331485

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

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

normal subgroups only

This group has 703 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,23)(19,22)(20,24)(25,26,31,37,30,34,29,33)(27,28,35,38,32,40,36,39);(2,5)(6,8)(9,11,12,18,13,19,20,17)(10,14,16,23,15,22,24,21)(25,27,31,35,30,32,29,36)(26,28,37,38,34,40,33,39);(3,7)(6,8)(11,17)(12,20)(14,21)(16,24)(18,19)(22,23)(25,28)(26,32)(27,34)(29,38)(30,40)(31,39)(33,36)(35,37);(1,3,4,7)(2,6,5,8)(10,15)(14,22)(16,24)(21,23)(25,29,30,31)(26,33,34,37)(27,35,32,36)(28,38,40,39)

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,d^c=de,f^c=fh,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, 238681);