#220618

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
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	#61216	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127814);