#474845

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382041);