#222283

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
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×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^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,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, 129479);