#6343729

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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2661 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,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, 6250925);