#190146

order	512
nilpotency class	3
isoclinism	isoclinic class of #190107
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

859 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,25,35,40)(19,27,30,41,29,23,34,37)(20,31,33,45,32,24,38,39)(28,36,44,47,43,42,46,48);(17,19,22,29)(18,23,25,27)(20,28,32,43)(21,34,35,30)(24,36,31,42)(26,41,40,37)(33,46,38,44)(39,48,45,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,24,25,31)(19,28)(21,33)(22,32)(23,36,27,42)(26,39,40,45)(29,43)(30,44)(34,46)(35,38)(37,47,41,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,29)(23,27)(28,43)(30,34)(36,42)(37,41)(44,46)(47,48)

Polycyclic presentation:

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

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, 97342);