#115221

order	512
nilpotency class	3
isoclinism	isoclinic class of #115213
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2404	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

901 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22417);