#316193

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

428 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:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=e,c^2=ef,d^2=eh,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, 223389);