#220204

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×#466	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

846 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:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(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);(3,9)(6,12)(10,15)(13,16)(17,19,24,33)(18,25,28,42)(20,35,21,30)(22,39,23,32)(26,44,34,40)(27,43,38,29)(31,36,48,37)(41,46,47,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(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);(3,9)(6,12)(10,15)(13,16)(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=h,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, 127400);