#180024

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

668 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)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,26,28)(33,34)(35,38)(36,39)(37,41)(40,44)(42,46)(43,45)(47,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,22,27)(18,23,20,26)(21,30,31,28)(24,32,29,25)(33,35)(34,38)(36,43)(37,42)(39,45)(40,47)(41,46)(44,48);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,20,22,18)(19,26,27,23)(21,24,31,29)(25,30,32,28)(34,39)(36,44)(38,45)(40,41)(43,48)(46,47);(2,8)(6,13)(7,14)(12,15)(18,20)(19,27)(24,29)(28,30)(35,42)(38,46)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,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, 87220);