#121910

order	512
nilpotency class	3
isoclinism	isoclinic class of #121909
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

384 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=i,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dhi,e^c=ehi,f^c=fhi,g^c=gi

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, 29106);