#193012

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1035 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)(4,7)(5,8)(9,10,14,18,15,19,13,27)(11,20,22,16,23,30,28,31)(12,17,24,33,26,34,25,32)(21,35,37,29,38,39,36,40);(2,6)(3,4)(7,8)(9,11,14,22,15,23,13,28)(10,16,18,30,19,31,27,20)(12,21,24,37,26,38,25,36)(17,29,33,39,34,40,32,35);(2,7)(3,4)(6,8)(9,12)(10,17)(11,21)(13,25)(14,24)(15,26)(16,29)(18,33)(19,34)(20,35)(22,37)(23,38)(27,32)(28,36)(30,39)(31,40);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,26,25)(16,20,31,30)(17,32,34,33)(21,36,38,37)(29,39,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100208);