#6343244

order	512
nilpotency class	3
isoclinism	isoclinic class of #62253
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62781	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2870 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250440);