#477533

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384729);