#6457707

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364903);