#118772

order	512
nilpotency class	3
isoclinism	isoclinic class of #118771
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

520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,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, 25968);