#209126

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#537	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

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

Polycyclic presentation:

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

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, 116322);