#279210

order	512
nilpotency class	3
isoclinism	isoclinic class of #279179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186406);