#496234

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#35088	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1789 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311488, #72311502, #72311506, …
maximal normal subgroup of	#72311488, #72311502, #72311506, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403430);