#178410

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

656 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)(3,11)(4,8)(5,14)(6,9)(7,12)(10,15)(13,16)(17,18,22,21)(19,23,26,29)(20,24,28,27)(25,31,32,30)(33,34,38,42)(35,43,46,47)(36,41,37,40)(39,44,48,45);(1,3,6,4)(2,7,9,16)(5,13,15,12)(8,14,11,10)(17,19,22,26)(18,23,21,29)(20,25,28,32)(24,30,27,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(1,4,6,3)(2,8,9,11)(5,13,15,12)(7,14,16,10)(17,20)(18,24)(19,25)(21,27)(22,28)(23,30)(26,32)(29,31)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45);(2,9)(3,4)(10,14)(12,13)(17,21,22,18)(19,23,26,29)(20,27,28,24)(25,30,32,31)(33,36,38,37)(34,41,42,40)(35,44,46,45)(39,43,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=h,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, 85606);