#116516

order	512
nilpotency class	3
isoclinism	isoclinic class of #116511
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

537 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)(19,29)(20,33,36,25)(21,28)(22,26)(23,27)(24,31)(30,45,47,41)(32,43)(34,48,46,39)(35,42,38,44)(37,40);(1,3,4,12,6,14,5,13)(2,7,9,8,11,15,10,16)(17,19,21,31,23,32,22,37)(18,24,26,29,27,40,28,43)(20,34,35,30,36,46,38,47)(25,41,42,39,33,45,44,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,10)(9,15)(11,16)(17,20,23,36)(18,25,27,33)(19,30,32,47)(21,35,22,38)(24,39,40,48)(26,42,28,44)(29,45,43,41)(31,46,37,34)

Polycyclic presentation:

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

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, 23712);