#117222

order	512
nilpotency class	3
isoclinism	isoclinic class of #117213
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^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=ei,f^b=fh,d^c=dh,e^c=ehi,g^c=gh

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, 24418);