#170405

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,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, 77601);