#187120

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	#62810	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2178 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,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, 94316);