#172673

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	#2679	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,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, 79869);