#172502

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=h,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, 79698);