#172648

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
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	#46090	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

756 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,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, 79844);