#172763

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
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

649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79959);