#328510

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37626	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 881 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,22)(14,19)(15,20)(17,21)(23,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(2,5)(6,8)(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,23)(22,24)(27,36)(28,29)(31,39)(32,33)(35,40)(37,38);(3,7)(6,8)(9,11)(10,15)(12,19)(13,23)(14,17)(16,24)(18,22)(20,21)(25,27,30,38)(26,31,34,40)(28,36,29,37)(32,35,33,39);(1,3,4,7)(2,6,5,8)(11,19)(14,18)(15,24)(20,23)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,35,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235706);