#178214

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

659 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dgh,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, 85410);