#322001

order	512
nilpotency class	3
isoclinism	isoclinic class of #321983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 718 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229197);