#415727

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
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

677 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:

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

Polycyclic presentation:

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

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, 322923);