#220217

order	512
nilpotency class	3
isoclinism	isoclinic class of #220210
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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,d^c=di,g^c=gh,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127413);