#235372

order	512
nilpotency class	3
isoclinism	isoclinic class of #235369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,e^c=ei,f^c=fhi

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, 142568);