#179208

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

669 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,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, 86404);