#345569

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
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

522 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 150 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)(9,10,14,16,13,18,12,17)(11,19,22,15,21,24,20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,39)(37,38);(1,3,5,8)(2,6,7,4)(27,34)(29,36)(30,38)(32,39)(33,40)(35,37);(9,11,13,21)(10,15,18,23)(12,22,14,20)(16,19,17,24)(25,27,28,35)(26,30,31,33)(29,37,36,34)(32,40,39,38);(1,4,5,6)(2,3,7,8)(10,16)(12,14)(15,19)(17,18)(20,22)(23,24)(26,31)(30,33)(32,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252765);