#179650

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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	#53347	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

750 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:

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

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86846);