#465972

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

883 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:

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

Polycyclic presentation:

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

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, 373168);