#6343423

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	C₂²×C₄	C₂⁵
prime radical	#62816	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2883 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,28,32,24)(20,33)(21,26)(22,25)(23,27)(29,37,31,42)(30,43,45,38)(34,40)(35,41)(36,39)(44,47,46,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,37)(26,38)(27,28)(33,42)(34,44)(35,45)(36,46)(39,47)(40,48)(41,43);(2,7)(6,14)(8,15)(9,16)(17,20,23,22)(18,25,27,33)(19,29,32,31)(21,34,35,36)(24,37,28,42)(26,39,41,40)(30,44,45,46)(38,47,43,48);(17,21,23,35)(18,26,27,41)(19,30,32,45)(20,34,22,36)(24,38,28,43)(25,39,33,40)(29,44,31,46)(37,47,42,48);(17,22,23,20)(18,25,27,33)(19,31,32,29)(21,34,35,36)(24,37,28,42)(26,40,41,39)(30,44,45,46)(38,48,43,47)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=i,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, 6250619);