#316197

order	512
nilpotency class	3
isoclinism	isoclinic class of #316187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

415 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)(3,5)(4,11)(6,10)(7,14)(8,12)(9,15)(13,16)(17,18,22,39,23,28,40,27)(19,20,30,24,31,34,25,33)(21,35,37,26,38,48,45,44)(29,43,46,32,36,41,42,47);(1,3,5,2)(4,7,13,9)(6,10,12,8)(11,15,16,14)(17,19,22,30,23,31,40,25)(18,24,27,20,28,33,39,34)(21,36,37,42,38,29,45,46)(26,43,35,47,44,41,48,32);(1,4,12,15,5,13,6,14)(2,7,10,16,3,9,8,11)(17,20,22,33,23,34,40,24)(18,25,39,31,28,30,27,19)(21,32,37,43,38,47,45,41)(26,42,48,36,44,46,35,29);(2,3)(7,9)(8,10)(11,16)(17,21,22,37,23,38,40,45)(18,26,27,35,28,44,39,48)(19,29,30,46,31,36,25,42)(20,32,33,43,34,47,24,41)

Polycyclic presentation:

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

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, 223393);