#180737

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

672 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31)(33,34,37,39)(35,41,42,48)(36,44,45,40)(38,46,47,43);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32)(36,45)(40,44)(41,48)(43,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29)(33,35,37,42)(34,38,39,47)(36,46,45,43)(40,48,44,41);(2,8)(3,10)(7,14)(9,16)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,27,32,31)(34,39)(38,47)(40,44)(41,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=g,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, 87933);