#190331

order	512
nilpotency class	3
isoclinism	isoclinic class of #190224
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fi,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, 97527);