#117432

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

525 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:

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

Polycyclic presentation:

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

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, 24628);