#493885

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1834 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22,23,32)(13,25,27,17)(14,18)(16,24)(19,33,34,40)(20,36,38,29)(21,30)(26,31)(28,35)(37,39);(2,5)(6,8)(9,11,14,21)(10,15,18,30)(12,19,23,34)(13,20,27,38)(16,28,31,39)(17,29,25,36)(22,33,32,40)(24,35,26,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,32)(18,31)(20,37)(21,34)(22,25)(24,27)(29,40)(30,39)(33,36)(35,38);(12,23)(16,31)(19,34)(22,32)(24,26)(28,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311869, #72311875, #72311883, …
maximal normal subgroup of	#72311869, #72311875, #72311883, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401081);