#187114

order	512
nilpotency class	3
isoclinism	isoclinic class of #186967
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,10,6,9,13,16)(3,8,11,15,4,7,12,14)(17,18)(19,27)(20,31,33,24)(21,34)(22,25)(23,29)(26,37)(28,43,45,48)(30,41)(32,40,36,42)(35,39)(38,44,47,46);(1,3,6,4)(2,7,9,8)(5,12,13,11)(10,15,16,14)(17,19,22,30)(18,23,25,39)(20,28,33,45)(21,35,37,29)(24,38,31,47)(26,41,34,27)(32,46,36,44)(40,43,42,48);(1,4,6,3)(2,8,9,7)(5,11,13,12)(10,14,16,15)(17,20,22,33)(18,24,25,31)(19,28,30,45)(21,36,37,32)(23,38,39,47)(26,42,34,40)(27,43,41,48)(29,46,35,44);(2,9)(7,8)(10,16)(14,15)(18,25)(23,39)(24,31)(26,34)(27,41)(38,47)(40,42)(43,48)

Polycyclic presentation:

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

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, 94310);