#190301

order	512
nilpotency class	3
isoclinism	isoclinic class of #190234
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

804 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)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,22,28,23,26,21,27)(19,29,32,41,31,24,36,39)(20,33,34,40,35,42,37,25)(30,44,47,43,45,48,46,38);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,23,31)(18,24,26,29)(20,30,35,45)(21,32,22,36)(25,38,40,43)(27,39,28,41)(33,48,42,44)(34,46,37,47);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25,26,40)(19,30)(21,34)(22,37)(23,35)(24,38,29,43)(27,33,28,42)(31,45)(32,46)(36,47)(39,48,41,44);(3,10)(6,14)(11,16)(13,15)(19,31)(24,29)(30,45)(32,36)(38,43)(39,41)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,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, 97497);