#420120

order	512
nilpotency class	3
isoclinism	isoclinic class of #417696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1180 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)(6,7)(9,10)(11,16)(12,25,27,34)(13,17)(14,19)(15,20)(18,26,32,28)(21,33)(22,36,38,39)(23,31)(24,29)(30,35,40,37);(1,2,4,5)(3,7,8,6)(9,11)(10,16)(12,22)(13,23)(14,21)(15,24)(17,31)(18,30)(19,33)(20,29)(25,36)(26,35)(27,38)(28,37)(32,40)(34,39);(10,17)(11,21)(14,15)(16,29)(18,32)(22,35)(23,24)(26,28)(30,39)(31,33)(36,40)(37,38);(1,3,4,8)(2,6,5,7)(9,12)(10,18)(11,22)(13,27)(14,26)(15,28)(16,30)(17,32)(19,25)(20,34)(21,35)(23,38)(24,37)(29,39)(31,40)(33,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=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, 327316);