#409649

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

824 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:

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

Polycyclic presentation:

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

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, 316845);