#190387

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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

824 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=h,f^2=hi,g^2=hi,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, 97583);