#492590

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33850	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309977
maximal normal subgroup of	#72309977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399786);