#117660

order	512
nilpotency class	3
isoclinism	isoclinic class of #117645
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

563 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24856);