#265214

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 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)(4,8)(5,9)(6,10)(7,13)(12,15)(14,16)(17,18,24,28)(19,29,33,31)(20,26,37,43)(21,22,27,23)(25,38,41,32)(30,42,47,45)(34,40,46,48)(35,36,44,39);(1,2)(3,12)(4,8)(5,9)(6,10)(7,14)(11,15)(13,16)(17,19,23,38,24,33,22,32)(18,25,27,31,28,41,21,29)(20,34,39,47,37,46,36,30)(26,42,44,48,43,45,35,40);(1,3,6,15)(2,7,10,16)(4,14,5,13)(8,11,9,12)(17,20)(18,26)(19,30)(21,35)(22,36)(23,39)(24,37)(25,40)(27,44)(28,43)(29,45)(31,42)(32,46)(33,47)(34,38)(41,48);(1,4,6,5)(2,8,10,9)(3,13,15,14)(7,12,16,11)(17,21,22,28,24,27,23,18)(19,31,32,25,33,29,38,41)(20,35,36,43,37,44,39,26)(30,42,46,40,47,45,34,48)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,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, 172410);