#441118

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,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, 348314);