#116671

order	512
nilpotency class	3
isoclinism	isoclinic class of #116666
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

540 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:

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

Polycyclic presentation:

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

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, 23867);