#187218

order	512
nilpotency class	3
isoclinism	isoclinic class of #187213
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2054 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)(3,10,7,4)(5,13)(6,9)(8,15)(11,12,14,16)(17,18)(19,27,30,41)(20,24)(21,26)(22,25)(23,29,37,33)(28,44,46,43)(31,39)(32,38)(34,42)(35,40)(36,45,48,47);(1,3,6,7)(2,4,9,10)(5,14,15,11)(8,16,13,12)(17,19)(18,23)(20,28)(21,33)(22,30)(24,36)(25,37)(26,41)(27,40)(29,35)(31,47)(32,46)(34,45)(38,48)(39,43)(42,44);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,13)(15,16)(17,20,22,32)(18,24,25,38)(19,28,30,46)(21,34,35,31)(23,36,37,48)(26,42,40,39)(27,43,41,44)(29,47,33,45);(1,5,6,15)(2,8,9,13)(3,11,7,14)(4,12,10,16)(18,25)(23,37)(24,38)(26,40)(27,41)(36,48)(39,42)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,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, 94414);