#6457713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

1954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=f,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,e^c=ei,e^d=ei

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, 6364909);