#186941

order	512
nilpotency class	3
isoclinism	isoclinic class of #186808
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	#62793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2217 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)(3,10,13,7)(4,8)(5,14)(6,9)(11,16,12,15)(17,18,22,27,23,26,21,34)(19,28,30,24,31,39,35,38)(20,25,32,42,33,40,36,41)(29,44,47,48,46,43,45,37);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,37)(26,38)(27,39)(28,34)(32,47)(33,46)(36,45)(40,48)(41,43)(42,44);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,23,33)(18,25,26,40)(19,29,31,46)(21,36,22,32)(24,37,38,48)(27,42,34,41)(28,43,39,44)(30,47,35,45);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,26)(24,38)(25,40)(27,34)(28,39)(37,48)(41,42)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94137);