#220702

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	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=i,e^2=h,f^2=h,g^2=hi,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, 127898);