#180751

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

606 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)(3,9)(4,6)(5,13)(7,11)(8,15)(10,14)(12,16)(20,27)(22,29)(23,28)(25,31)(33,34,39,44)(35,45,41,48)(36,47,40,46)(37,42,38,43);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(17,18)(19,22)(20,28)(21,24)(23,27)(25,32)(26,29)(30,31)(33,35,39,41)(34,40,44,36)(37,46,38,47)(42,45,43,48);(1,4,6,2)(3,10,12,7)(5,13,15,8)(9,11,16,14)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,30,28,32)(33,36,39,40)(34,41,44,35)(37,45,38,48)(42,46,43,47);(2,4)(7,10)(8,13)(9,16)(17,20,21,27)(18,23,24,28)(19,25,26,31)(22,30,29,32)(33,37,39,38)(34,42,44,43)(35,46,41,47)(36,48,40,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,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, 87947);