#6343677

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343394
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	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6250873);