#186485

order	512
nilpotency class	3
isoclinism	isoclinic class of #186176
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2015 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93681);