#118298

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
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×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25494);