#174576

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
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

720 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:

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

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81772);