#119769

order	512
nilpotency class	3
isoclinism	isoclinic class of #119765
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fi,g^b=gh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26965);