#118265

order	512
nilpotency class	3
isoclinism	isoclinic class of #9986
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh

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, 25461);