#6362577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#18578	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1553 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,22,25)(19,27,31,40)(20,24,33,38)(21,34,35,26)(23,29,37,30)(28,41,44,48)(32,45,46,39)(36,42,47,43);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,22,31)(18,23,25,37)(20,28,33,44)(21,29,35,30)(24,36,38,47)(26,40,34,27)(32,42,46,43)(39,48,45,41);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,20,22,33)(18,24,25,38)(19,28,31,44)(21,32,35,46)(23,36,37,47)(26,39,34,45)(27,41,40,48)(29,42,30,43);(18,25)(19,29)(21,35)(23,27)(24,38)(28,42)(30,31)(32,46)(36,41)(37,40)(43,44)(47,48);(2,9)(7,13)(8,16)(11,15)(18,25)(23,37)(24,38)(26,34)(27,40)(36,47)(39,45)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269773);