#187708

order	512
nilpotency class	3
isoclinism	isoclinic class of #187667
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62752	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2039 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)(3,11)(4,13)(5,9)(6,16)(7,10)(8,12)(14,15)(17,18)(19,29,32,24)(20,33)(21,28)(22,26)(23,27)(25,34)(30,44,46,48)(31,39,36,41)(35,42)(37,40)(38,47,43,45);(1,3)(2,8)(4,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,38)(26,39)(27,29)(28,41)(33,44)(34,45)(35,46)(37,47)(40,43)(42,48);(1,4,7,8)(2,3,10,14)(5,11,6,15)(9,12,16,13)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,33)(27,40)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(1,5,7,6)(2,9,10,16)(3,12,14,13)(4,11,8,15)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)

Polycyclic presentation:

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

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, 94904);