#189324

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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(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)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,21,32,23,33,22,31)(18,24,27,29,28,39,26,42)(20,30,35,45,36,46,37,44)(25,38,40,43,41,48,34,47);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(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);(3,10)(6,14)(11,16)(13,15)(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=e,d^2=h,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, 96520);