#6343665

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	#62781	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2874 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18)(19,27,30,23)(20,31)(21,34)(22,26)(24,32)(25,35)(28,36,41,39)(29,37,43,40)(33,38)(42,47,48,46)(44,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,37)(26,27)(31,39)(32,41)(33,42)(34,40)(35,43)(38,46)(44,47)(45,48);(2,8)(7,15)(9,13)(10,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,38)(26,31)(27,39)(29,42)(30,41)(34,44)(35,45)(37,46)(40,47)(43,48);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(20,32)(24,31)(28,41)(33,45)(36,39)(38,44)(42,48)(46,47);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,22,35)(18,25,26,34)(19,29,30,43)(20,33,32,45)(23,37,27,40)(24,38,31,44)(28,42,41,48)(36,46,39,47)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,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, 6250861);