#180664

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

664 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32)(33,34,37,41)(35,38,42,46)(36,39,44,40)(43,45,48,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,28,30,31)(33,35)(34,38)(36,43)(37,42)(39,45)(40,47)(41,46)(44,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)(34,39)(36,44)(38,45)(40,41)(43,48)(46,47);(2,7)(6,15)(8,12)(9,16)(18,24)(19,26)(23,30)(25,32)(35,42)(38,46)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^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, 87860);