#187424

order	512
nilpotency class	3
isoclinism	isoclinic class of #187419
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	#62840	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2065 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94620);