#341721

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

498 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=ehi,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248917);