#6351140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2608 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)(3,12)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(17,18,22,25)(19,27,30,40)(20,24,31,38)(21,33,35,26)(23,34,37,29)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19,22,30)(18,23,25,37)(20,28,31,42)(21,34,35,29)(24,36,38,47)(26,40,33,27)(32,46,44,43)(39,48,45,41);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,22,31)(18,24,25,38)(19,28,30,42)(21,32,35,44)(23,36,37,47)(26,39,33,45)(27,41,40,48)(29,43,34,46);(1,2)(3,8)(4,16)(5,10)(6,11)(7,9)(12,13)(14,15)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46);(2,9)(8,16)(10,11)(12,15)(18,25)(23,37)(24,38)(26,33)(27,40)(36,47)(39,45)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258336);