#220620

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61216	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127816);