#464714

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1520 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371910);