#6358551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1542 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)(17,18,22,25)(19,27,31,37)(20,24,33,39)(21,34,35,26)(23,29,38,30)(28,41,44,47)(32,45,46,40)(36,42,48,43);(17,19,22,31)(18,23,25,38)(20,28,33,44)(21,29,35,30)(24,36,39,48)(26,37,34,27)(32,42,46,43)(40,47,45,41);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,22,33)(18,24,25,39)(19,28,31,44)(21,32,35,46)(23,36,38,48)(26,40,34,45)(27,41,37,47)(29,42,30,43);(19,29)(21,35)(23,37)(26,34)(27,38)(28,42)(30,31)(32,46)(36,47)(40,45)(41,48)(43,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,25)(23,38)(24,39)(26,34)(27,37)(36,48)(40,45)(41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265747);