#330256

order	512
nilpotency class	3
isoclinism	isoclinic class of #329551
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

568 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,6)(4,5)(7,8)(9,10,13,16,14,15,22,23)(11,12,18,19,17,20,24,21)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(2,5)(6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(3,7)(6,8)(9,12,14,20)(10,11,15,17)(13,21,22,19)(16,24,23,18)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,23)(18,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,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, 237452);