#172776

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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

703 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)(17,18,22,21)(19,23,26,29)(20,24,28,27)(25,31,32,30)(33,34)(35,43)(36,41)(37,40)(38,42)(39,45)(44,48)(46,47);(1,3,7,14)(2,8,4,11)(5,13,6,12)(9,16,15,10)(17,19,22,26)(18,23,21,29)(20,25,28,32)(24,30,27,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(1,4,7,2)(3,11,14,8)(5,9,6,15)(10,13,16,12)(17,20,22,28)(18,24,21,27)(19,25,26,32)(23,30,29,31)(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)(17,21,22,18)(19,23,26,29)(20,27,28,24)(25,30,32,31)(33,36,38,37)(34,41,42,40)(35,44,46,45)(39,43,48,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=ghi,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, 79972);