#186940

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	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2227 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(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)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(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);(17,20)(18,25)(19,29)(21,36)(22,32)(23,33)(24,37)(26,40)(27,42)(28,43)(30,47)(31,46)(34,41)(35,45)(38,48)(39,44);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,26)(24,38)(25,40)(27,34)(28,39)(37,48)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=g,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, 94136);