#178412

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

652 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85608);