#184369

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ci,d^a=dh,e^a=eh,c^b=ci,e^b=eh,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91565);