#191233

order	512
nilpotency class	3
isoclinism	isoclinic class of #191231
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

525 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)(3,5)(4,12)(6,9)(7,13)(8,10)(11,14)(15,16)(17,18)(19,27,32,23)(20,33,35,48)(21,25)(22,26)(24,34,40,42)(28,36)(29,43)(30,39,31,38)(37,45)(41,44)(46,47);(1,3,10,9,6,8,5,2)(4,11,16,15,14,12,13,7)(17,19,28,30,22,32,21,31)(18,23,36,38,26,27,25,39)(20,29,42,44,35,46,34,45)(24,37,48,43,40,41,33,47);(1,4,5,13,6,14,10,16)(2,7,8,12,9,15,3,11)(17,20)(18,24,26,40)(19,29)(21,34)(22,35)(23,37,27,41)(25,33,36,48)(28,42)(30,44)(31,45)(32,46)(38,43,39,47);(1,5,6,10)(2,3,9,8)(4,13,14,16)(7,11,15,12)(17,21,22,28)(18,25,26,36)(19,30,32,31)(20,34,35,42)(23,38,27,39)(24,33,40,48)(29,44,46,45)(37,43,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98429);