#195077

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1982 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)(3,7,8,5)(4,6)(9,10)(11,20,23,32)(12,24)(13,19)(14,28)(15,18)(16,22,31,27)(17,26)(21,36,37,30)(25,33)(29,34)(35,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,27,15,23,14,22)(10,16,19,32,18,31,28,20)(12,21,26,39,25,37,29,38)(17,30,33,40,34,36,24,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,26)(14,29)(15,25)(16,30)(18,34)(19,33)(20,35)(22,38)(23,37)(24,28)(27,39)(31,36)(32,40);(10,18)(12,25)(16,31)(19,28)(20,32)(21,37)(26,29)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102273);