#219517

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#23810	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efh,d^2=ehi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126713);