#118257

order	512
nilpotency class	3
isoclinism	isoclinic class of #10560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 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:

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

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,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25453);