#220606

order	512
nilpotency class	3
isoclinism	isoclinic class of #220528
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

758 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127802);