#187569

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	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2152 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)(3,11,13,7,14,15,12,16)(17,18)(19,28,32,24)(20,33,36,41)(21,37)(22,26)(23,27)(25,35,40,34)(29,44)(30,39,31,43)(38,46)(42,45)(47,48);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(17,19,23,32)(18,24,27,28)(20,29,36,47)(21,30,22,31)(25,38,40,42)(26,39,37,43)(33,48,41,44)(34,45,35,46);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20)(18,25)(19,29)(21,35)(22,34)(23,36)(24,38)(26,41)(27,40)(28,42)(30,46)(31,45)(32,47)(33,37)(39,44)(43,48);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,11,16,15)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,39,28,43)(25,33,40,41)(29,45,47,46)(38,48,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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, 94765);