#187885

order	512
nilpotency class	3
isoclinism	isoclinic class of #187866
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	#62755	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2066 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)(3,11)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(17,18)(19,28,31,24)(20,32,34,41)(21,35)(22,26)(23,27)(25,33,42,37)(29,44)(30,39,36,40)(38,47)(43,45)(46,48);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,38)(26,39)(27,28)(32,44)(33,45)(34,46)(35,40)(37,47)(41,48)(42,43);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,35)(36,47)(39,48)(40,44);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,34,37)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44)

Polycyclic presentation:

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

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, 95081);