#331498

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

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

644 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 154 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)(9,10,15,17,14,18,13,23)(11,12,19,22,20,16,24,21)(25,26)(27,33)(28,34)(29,31)(30,32)(35,38)(36,40)(37,39);(2,5)(6,8)(9,11)(10,16)(12,18)(13,24)(14,20)(15,19)(17,21)(22,23)(25,27,30,35,29,34,37,40)(26,28,32,36,31,33,39,38);(3,7)(6,8)(9,12,14,16)(10,11,18,20)(13,22,15,21)(17,24,23,19)(25,28)(26,27)(29,33)(30,38)(31,34)(32,40)(35,39)(36,37);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,17,18,23)(11,19,20,24)(12,21,16,22)(26,31)(28,33)(32,39)(36,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,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, 238694);