#117442

order	512
nilpotency class	3
isoclinism	isoclinic class of #117433
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

533 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,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=fh,c^b=cf,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,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, 24638);