#186164

order	512
nilpotency class	3
isoclinism	isoclinic class of #186162
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	#62827	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2063 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=di,f^c=fh,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, 93360);