#188927

order	512
nilpotency class	3
isoclinism	isoclinic class of #188818
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

879 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96123);