#187863

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62755	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2083 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)(19,29,32,24)(20,33,35,42)(21,26)(22,28)(23,27)(25,34,40,37)(30,44)(31,41,36,39)(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)(20,30)(21,31)(22,36)(23,32)(25,38)(26,39)(27,29)(28,41)(33,44)(34,45)(35,46)(37,47)(40,43)(42,48);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,33)(27,40)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95059);