#174993

order	512
nilpotency class	3
isoclinism	isoclinic class of #174829
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

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,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, 82189);