#254203

order	512
nilpotency class	3
isoclinism	isoclinic class of #254179
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	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,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,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

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, 161399);