#474057

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	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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)(9,10,16,11,14,19,15,20)(12,23,25,21,26,17,30,33)(13,27,31,34,28,18,29,22)(24,36,38,35,37,32,39,40);(2,5)(6,8)(9,11,16,19,14,20,15,10)(12,17,25,33,26,23,30,21)(13,22,31,27,28,34,29,18)(24,32,38,40,37,36,39,35);(1,3,4,7)(2,6,5,8)(9,12)(10,17,19,23)(11,21,20,33)(13,24,28,37)(14,26)(15,25)(16,30)(18,32)(22,35)(27,36)(29,38,31,39)(34,40);(3,7)(6,8)(9,13)(10,18)(11,22)(12,24)(14,28)(15,29)(16,31)(17,32)(19,27)(20,34)(21,35)(23,36)(25,38)(26,37)(30,39)(33,40)

Polycyclic presentation:

2^9|a^2=eg,b^2=gh,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 381253);