#176712

order	512
nilpotency class	3
isoclinism	isoclinic class of #176333
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

628 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83908);