#6343725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
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	#62831	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2710 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

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, 6250921);