#187859

order	512
nilpotency class	3
isoclinism	isoclinic class of #187780
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×#3562	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95055);