#127246

order	512
nilpotency class	4
isoclinism	isoclinic class of #127217
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,21,26,23,27,22,36)(19,28,30,44,32,45,31,48)(20,25,34,42,33,41,35,43)(24,37,39,29,38,47,40,46);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19,22,31,23,32,21,30)(18,24,26,40,27,38,36,39)(20,29,34,47,33,46,35,37)(25,28,43,44,41,45,42,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(20,33)(24,38)(25,41)(29,46)(34,35)(37,47)(39,40)(42,43)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgi,e^b=eh,d^c=di,e^d=ei

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, 34442);