#409255

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

689 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)(11,20,23,29)(12,17)(13,18)(14,28)(15,19)(16,21,31,24)(22,34,37,39)(25,32)(26,33)(27,38)(30,35,40,36);(1,3,4,7)(2,5,6,8)(9,11,13,24,15,23,14,21)(10,16,18,29,19,31,28,20)(12,22,26,36,25,37,27,35)(17,30,33,39,32,40,38,34);(11,21,23,24)(12,25)(16,29,31,20)(17,32)(22,36,37,35)(26,27)(30,34,40,39)(33,38);(2,6)(5,8)(9,12,13,26,15,25,14,27)(10,17,18,33,19,32,28,38)(11,22,24,36,23,37,21,35)(16,30,29,39,31,40,20,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316451);