#118485

order	512
nilpotency class	3
isoclinism	isoclinic class of #118478
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

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:

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

Polycyclic presentation:

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

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, 25681);