#117989

order	512
nilpotency class	3
isoclinism	isoclinic class of #117987
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

546 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)(3,9,11,6)(4,12)(5,7)(8,14)(10,15,13,16)(17,18,22,26,23,28,21,27)(19,29,32,40,33,44,31,24)(20,34,37,25,38,42,36,43)(30,39,46,45,47,48,35,41);(3,10,11,13)(6,15,9,16)(17,19,21,31,23,33,22,32)(18,24,26,29,28,40,27,44)(20,35,36,47,38,46,37,30)(25,41,42,39,43,45,34,48);(1,3)(2,6)(4,13)(5,11)(7,9)(8,16)(10,14)(12,15)(17,20,23,38)(18,25,28,43)(19,30,33,47)(21,36,22,37)(24,39,40,48)(26,42,27,34)(29,45,44,41)(31,35,32,46)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=eh,f^c=fhi,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, 25185);