#6344116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
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	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2795 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=i,g^2=hi,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=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, 6251312);