#6344206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343524
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	#62836	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2823 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bf,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=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, 6251402);