#189329

order	512
nilpotency class	3
isoclinism	isoclinic class of #189323
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(17,18)(19,29)(20,34,36,40)(21,26)(22,27)(23,28)(24,32)(25,35,41,37)(30,38,46,48)(31,39)(33,42)(43,45,47,44);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,22,31,23,33,21,32)(18,24,26,42,28,39,27,29)(20,30,37,44,36,46,35,45)(25,38,34,47,41,48,40,43);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,40)(28,41)(29,43)(31,44)(32,45)(33,46)(39,48)(42,47);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,29,39,42)(25,34,41,40)(30,44,46,45)(38,43,48,47)

Polycyclic presentation:

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

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, 96525);