#316206

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

Polycyclic presentation:

2^9|a^2=egi,b^2=e,c^2=efi,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, 223402);