#254271

order	512
nilpotency class	3
isoclinism	isoclinic class of #254227
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

813 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 813 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

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, 161467);