#215006

order	512
nilpotency class	3
isoclinism	isoclinic class of #214736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

880 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cei,f^b=fh,g^b=gh,f^c=fi,g^d=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, 122202);