#493882

order	512
nilpotency class	4
isoclinism	isoclinic class of #480250
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1786 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)(12,17)(13,24,25,16)(14,18)(19,27)(20,34,35,26)(21,28)(22,31)(23,30,36,29)(32,39)(33,38,40,37);(2,5)(6,8)(9,11,14,21)(10,15,18,28)(12,19,22,32)(13,20,25,35)(16,26,24,34)(17,27,31,39)(23,33,36,40)(29,37,30,38);(1,3,4,7)(2,6,5,8)(10,16)(12,22)(15,26)(17,30)(18,24)(19,32)(23,36)(27,38)(28,34)(29,31)(33,40)(37,39);(9,12,14,22)(10,17,18,31)(11,19,21,32)(13,23,25,36)(15,27,28,39)(16,29,24,30)(20,33,35,40)(26,37,34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306422, #72306425, #72306427
maximal normal subgroup of	#72306422, #72306425, #72306427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401078);