#331478

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
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

668 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 158 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)(9,10,14,17,13,16,21,24)(11,12,19,20,18,15,23,22)(25,26,31,35,30,33,29,27)(28,36,39,32,38,34,37,40);(2,5)(6,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,24)(21,23)(25,27,29,33,30,35,31,26)(28,32,37,36,38,40,39,34);(3,7)(6,8)(9,12,13,15)(10,11,16,18)(14,22,21,20)(17,23,24,19)(25,28,30,38)(26,32,33,40)(27,34,35,36)(29,39,31,37);(1,3,4,7)(2,6,5,8)(11,18)(12,15)(19,23)(20,22)(26,33)(27,35)(32,40)(34,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,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, 238674);