#466293

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=fi,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, 373489);