#117690

order	512
nilpotency class	3
isoclinism	isoclinic class of #117681
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

575 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)(3,11)(4,9)(5,8)(6,10)(7,13)(12,15)(14,16)(17,18,22,26,23,27,21,38)(19,28,32,40,31,44,30,24)(20,33,36,42,37,25,35,41)(29,45,47,48,34,43,46,39);(1,3,4,13,6,12,5,14)(2,7,8,11,10,16,9,15)(17,19,21,30,23,31,22,32)(18,24,26,28,27,40,38,44)(20,34,35,47,37,29,36,46)(25,39,41,45,33,48,42,43);(2,8)(3,12)(4,5)(7,15)(9,10)(11,16)(17,20,23,37)(18,25,27,33)(19,29,31,34)(21,36,22,35)(24,39,40,48)(26,42,38,41)(28,43,44,45)(30,47,32,46)

Polycyclic presentation:

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

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, 24886);