#6344156

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251352);