#6352452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352341
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62881	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2706 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)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,26)(19,27,30,40)(20,24,31,38)(21,33,35,25)(23,34,37,29)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,22,30)(18,23,26,37)(20,28,31,42)(21,34,35,29)(24,36,38,47)(25,40,33,27)(32,46,44,43)(39,48,45,41);(1,2,6,9)(3,7,12,16)(4,8,5,13)(10,11,15,14)(17,20,22,31)(18,24,26,38)(19,28,30,42)(21,32,35,44)(23,36,37,47)(25,39,33,45)(27,41,40,48)(29,43,34,46);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259648);