#497519

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#547	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

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,c^b=ci,d^b=de,g^c=gh,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404715);