#173579

order	512
nilpotency class	3
isoclinism	isoclinic class of #173481
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgh,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, 80775);