#170369

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77565);