#6344132

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343418
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	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2833 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251328);