#200575

order	512
nilpotency class	3
isoclinism	isoclinic class of #200418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1292 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,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, 107771);