#187112

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	C₂×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(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);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(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);(2,7)(4,13)(6,9)(10,12)(17,20)(18,24)(19,28)(21,36)(22,33)(23,38)(25,31)(26,42)(27,43)(29,46)(30,45)(32,37)(34,40)(35,44)(39,47)(41,48);(2,7)(6,9)(8,15)(14,16)(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,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, 94308);