#185608

order	512
nilpotency class	3
isoclinism	isoclinic class of #185517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2011 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:

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

Polycyclic presentation:

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

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, 92804);