#115529

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22725);