#408254

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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1091 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)(11,16)(12,26,29,37)(13,25)(14,18)(15,20)(17,30,36,32)(19,21)(22,35,38,40)(23,34)(24,31)(27,33,39,28);(1,3,4,7)(2,5,6,8)(9,11,21,24,15,25,14,23)(10,16,19,31,20,13,18,34)(12,27,30,22,29,39,32,38)(17,35,26,28,36,40,37,33);(9,12)(10,17)(11,22)(13,28)(14,32)(15,29)(16,33)(18,37)(19,26)(20,36)(21,30)(23,27)(24,39)(25,38)(31,35)(34,40);(2,6)(5,8)(9,13,14,31,15,16,21,34)(10,11,18,23,20,25,19,24)(12,28,32,35,29,33,30,40)(17,22,37,27,36,38,26,39)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,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, 315450);