#265220

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18,24,28)(19,29,32,37)(20,26,36,44)(21,23,27,22)(25,38,41,31)(30,42,47,45)(33,40,46,48)(34,39,43,35);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(17,19,22,31,24,32,23,38)(18,25,27,29,28,41,21,37)(20,33,35,30,36,46,39,47)(26,42,43,40,44,45,34,48);(1,3,4,13)(2,8,7,15)(5,12,6,14)(9,16,10,11)(17,20)(18,26)(19,30)(21,34)(22,35)(23,39)(24,36)(25,40)(27,43)(28,44)(29,45)(31,46)(32,47)(33,38)(37,42)(41,48);(2,7)(3,13)(9,10)(12,14)(17,21,22,18,24,27,23,28)(19,29,31,41,32,37,38,25)(20,34,35,26,36,43,39,44)(30,45,46,48,47,42,33,40)

Polycyclic presentation:

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

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, 172416);