#451894

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
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	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=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, 359090);