#220209

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
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

830 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127405);