#440971

order	512
nilpotency class	3
isoclinism	isoclinic class of #419445
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1100 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=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, 348167);