#6343737

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	#62823	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2720 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,28,32,24)(20,25)(21,35)(22,26)(23,27)(29,42,45,37)(30,38,31,43)(33,40)(34,41)(36,39)(44,47,46,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,32)(18,24,27,28)(20,29,34,45)(21,30,22,31)(25,37,41,42)(26,38,35,43)(33,44,36,46)(39,47,40,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,39)(27,41)(28,42)(30,44)(31,46)(32,45)(35,40)(38,47)(43,48);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,27,35)(19,30,32,31)(20,33,34,36)(24,38,28,43)(25,39,41,40)(29,44,45,46)(37,47,42,48);(2,7)(6,15)(8,12)(9,16)(17,22,23,21)(18,26,27,35)(19,31,32,30)(20,33,34,36)(24,38,28,43)(25,40,41,39)(29,44,45,46)(37,48,42,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=hi,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, 6250933);