#180650

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
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	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87846);