#118394

order	512
nilpotency class	3
isoclinism	isoclinic class of #118389
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,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,f^b=fhi,d^c=di,f^c=fh,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, 25590);