#187574

order	512
nilpotency class	3
isoclinism	isoclinic class of #187442
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	#62830	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2182 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)(3,11,14,7)(4,15)(5,9)(6,10)(8,13,16,12)(17,18,22,26,23,28,21,27)(19,29,31,24,32,39,37,40)(20,33,36,25,34,42,35,41)(30,44,46,48,45,38,47,43);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,11)(15,16)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,38)(26,39)(27,29)(28,40)(33,48)(34,45)(35,47)(36,46)(41,44)(42,43);(1,3)(2,8)(4,13)(5,12)(6,14)(7,9)(10,16)(11,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,38)(26,33)(27,42)(28,41)(29,43)(31,47)(32,45)(37,46)(39,48)(40,44);(1,4,6,5)(2,9,10,15)(3,12,14,13)(7,8,11,16)(20,34)(25,41)(30,45)(33,42)(35,36)(38,44)(43,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94770);