#180460

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

640 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,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, 87656);