#458228

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

933 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)(3,6)(4,8)(5,7)(9,10)(11,21)(12,17)(13,28)(14,31)(15,20)(16,24)(18,30)(19,29)(22,36)(23,32)(25,38)(26,33)(27,35)(34,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11,14,24,15,25,29,40)(10,16,19,21,20,34,31,38)(12,26,13,23,27,22,30,39)(17,32,18,33,35,37,28,36);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,29)(14,30)(15,27)(16,32)(18,31)(19,28)(20,35)(21,36)(23,24)(25,26)(33,38)(34,37)(39,40);(9,13,29,12,15,30,14,27)(10,18,31,17,20,28,19,35)(11,23,40,26,25,39,24,22)(16,33,38,32,34,36,21,37)

Polycyclic presentation:

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

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, 365424);