#254078

order	512
nilpotency class	3
isoclinism	isoclinic class of #253987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

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, 161274);