#325336

order	512
nilpotency class	3
isoclinism	isoclinic class of #278011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

568 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 170 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)(4,6)(5,8)(9,10)(11,17)(12,14)(13,21)(15,20)(16,18)(19,23)(22,24)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,3,5,4)(2,6,8,7)(10,14)(15,22)(16,21)(17,23)(25,27,28,34,29,35,36,33)(26,30,31,38,32,39,40,37);(1,3,5,4)(2,7,8,6)(9,11,18,24,12,19,13,20)(10,15,21,23,14,22,16,17)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(1,4,5,3)(2,7,8,6)(10,14)(15,22)(16,21)(17,23)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232532);