#441635

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

877 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=efi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,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, 348831);