#410464

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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)(9,10,14,20,16,21,15,36)(11,22,25,38,27,37,26,17)(12,28,30,18,32,39,31,24)(13,19,33,40,35,23,34,29);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,29)(13,24)(14,26)(15,25)(16,27)(18,35)(19,31)(20,37)(21,38)(22,36)(23,30)(28,34)(32,40)(33,39);(9,12,15,30,16,32,14,31)(10,18,20,28,21,24,36,39)(11,23,26,29,27,19,25,40)(13,17,34,22,35,38,33,37);(2,6)(5,8)(9,13,14,33,16,35,15,34)(10,19,20,40,21,23,36,29)(11,24,26,39,27,18,25,28)(12,17,31,37,32,38,30,22)

Polycyclic presentation:

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

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, 317660);