#179359

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

623 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)(3,10)(4,7)(5,9)(6,15)(8,13)(11,12)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(33,34,38,42)(35,43,46,47)(36,41,37,40)(39,44,48,45);(1,3,7,14)(2,8,4,11)(5,13,6,12)(9,16,15,10)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31)(33,35,38,46)(34,39,42,48)(36,44,37,45)(40,47,41,43);(1,4,7,2)(3,11,14,8)(5,9,6,15)(10,13,16,12)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45);(1,5,7,6)(2,9,4,15)(3,12,14,13)(8,10,11,16)(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=g,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgi,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, 86555);