#499598

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,15)(13,25)(14,18)(17,23)(19,30)(20,31)(21,33)(22,28)(24,29)(26,34)(27,40)(32,37)(35,38)(36,39);(1,3,4,7)(2,5,6,8)(10,15)(11,19)(13,26)(14,24)(16,30)(17,34)(20,27)(21,38)(22,37)(23,39)(25,36)(28,32)(31,40)(33,35);(2,6)(5,8)(9,11)(10,16)(12,22)(13,27)(14,20)(15,28)(17,35)(18,31)(19,26)(21,24)(23,38)(25,40)(29,33)(30,34)(32,36)(37,39);(11,20,22,21)(13,23)(16,31,28,33)(17,25)(19,27,37,38)(26,39)(30,40,32,35)(34,36)

Polycyclic presentation:

2^9|b^2=f,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239832, #72239836, #72239840
maximal normal subgroup of	#72239832, #72239836, #72239840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406794);