#190156

order	512
nilpotency class	3
isoclinism	isoclinic class of #190124
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

936 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,g^b=gh,f^c=fh,g^c=gi

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, 97352);