#116854

order	512
nilpotency class	3
isoclinism	isoclinic class of #116849
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,f^b=fhi,e^c=ehi,f^c=fhi,g^c=gh

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, 24050);