#264020

order	512
nilpotency class	3
isoclinism	isoclinic class of #264003
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	#3558	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 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)(4,12)(5,8)(6,10)(7,14)(9,15)(13,16)(17,18,24,28)(19,29,33,31)(20,26,37,44)(21,23,27,22)(25,38,41,32)(30,42,47,45)(34,40,46,48)(35,39,43,36);(1,3,5,12)(2,4,8,11)(6,7,15,13)(9,14,10,16)(17,19,22,32,24,33,23,38)(18,25,27,29,28,41,21,31)(20,34,36,30,37,46,39,47)(26,42,43,40,44,45,35,48);(1,4,5,11)(2,7,8,13)(3,10,12,9)(6,16,15,14)(17,20)(18,26)(19,30)(21,35)(22,36)(23,39)(24,37)(25,40)(27,43)(28,44)(29,45)(31,42)(32,46)(33,47)(34,38)(41,48);(2,8)(4,11)(9,10)(14,16)(17,21,22,18,24,27,23,28)(19,31,32,25,33,29,38,41)(20,35,36,26,37,43,39,44)(30,42,46,40,47,45,34,48)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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=gh,c^b=cg,f^b=fh,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171216);