#426817

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 334013);