#330724

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

538 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237920);