#187473

order	512
nilpotency class	3
isoclinism	isoclinic class of #187410
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	#62823	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2042 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,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, 94669);