#409751

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

795 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316947);