#179223

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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	#45962	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

729 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)(3,11)(4,8)(5,14)(6,10)(7,12)(9,15)(13,16)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28)(33,34)(35,43)(36,39)(37,41)(38,42)(40,46)(44,47)(45,48);(1,3,6,4)(2,7,10,16)(5,13,15,12)(8,14,11,9)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31)(34,39)(36,37)(40,47)(41,42)(43,48)(45,46);(1,4,6,3)(2,8,10,11)(5,13,15,12)(7,14,16,9)(20,21)(24,25)(27,32)(28,30)(33,35)(34,40)(36,46)(37,45)(38,44)(39,47)(41,43)(42,48);(3,4)(7,16)(8,11)(12,13)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)(35,44)(40,48)(43,47)(45,46)

Polycyclic presentation:

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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86419);