#187453

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1994 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,29,33,24)(20,34,37,41)(21,26)(22,28)(23,27)(25,36,42,35)(30,44)(31,40,32,39)(38,46)(43,45)(47,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,33)(18,24,27,29)(20,30,37,47)(21,31,22,32)(25,38,42,43)(26,39,28,40)(34,48,41,44)(35,45,36,46);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,34)(27,42)(28,41)(29,43)(31,46)(32,45)(33,47)(39,48)(40,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,35,37,36)(24,39,29,40)(25,41,42,34)(30,45,47,46)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=di,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, 94649);