#235505

order	512
nilpotency class	3
isoclinism	isoclinic class of #235504
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	#37556	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142701);