#187511

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62748	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2193 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:

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

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,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,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, 94707);