#6344214

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	#62835	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bf,d^a=dh,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251410);