#294912

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,7)(4,5)(6,8)(9,10,12,15)(11,14,19,23)(13,21,20,16)(17,22,18,24)(25,26)(27,36)(28,39)(29,35)(30,34)(31,33)(32,38)(37,40);(2,5)(6,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,24)(18,20)(21,22)(25,27,30,38)(26,32,34,36)(28,31,37,29)(33,39,35,40);(10,15)(11,17,19,18)(14,22,23,24)(16,21)(25,28,31,38,30,37,29,27)(26,32,35,39,34,36,33,40);(1,3,4,8)(2,6,5,7)(25,29,30,31)(26,33,34,35)(27,37,38,28)(32,40,36,39)

Polycyclic presentation:

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

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, 202108);