#187579

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#3529	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2308 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=gh

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, 94775);