#418068

order	512
nilpotency class	3
isoclinism	isoclinic class of #417579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1485 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:

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

Polycyclic presentation:

2^9|c^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325264);