#6344134

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	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2809 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)(17,18,21,26,22,24,34,38)(19,27,29,40,30,23,33,36)(20,25,32,39,31,37,44,47)(28,41,43,48,42,35,45,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,19,22,30)(18,23,24,27)(20,28,31,42)(21,33,34,29)(25,35,37,41)(26,40,38,36)(32,45,44,43)(39,48,47,46);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,16,15)(18,24)(23,27)(25,37)(26,38)(35,41)(36,40)(39,47)(46,48);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,15,16,11)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,37)(26,39)(27,41)(29,43)(30,42)(33,45)(34,44)(36,46)(38,47)(40,48);(20,31)(25,37)(28,42)(32,44)(35,41)(39,47)(43,45)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,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, 6251330);