#312155

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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

562 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219351);