#200427

order	512
nilpotency class	3
isoclinism	isoclinic class of #200424
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35144	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1186 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)(3,7)(6,8)(9,10)(11,15,21,30)(12,22,24,16)(13,25)(14,18)(17,27)(19,28)(20,32,26,29)(23,31,38,36)(33,40)(34,35)(37,39);(1,2,3,5,4,6,7,8)(9,11)(10,15,18,30)(12,19)(13,26)(14,21)(16,28,22,35)(17,32,25,29)(20,27)(23,37)(24,34)(31,40,36,39)(33,38);(9,12)(10,16)(11,19)(13,23)(14,24)(15,28)(17,31)(18,22)(20,33)(21,34)(25,36)(26,37)(27,38)(29,39)(30,35)(32,40);(2,6)(5,8)(9,13,14,27)(10,17,18,25)(11,20,21,26)(12,23,24,38)(15,29,30,32)(16,31,22,36)(19,33,34,37)(28,39,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107623);