#174913

order	512
nilpotency class	3
isoclinism	isoclinic class of #174829
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	#38433	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

873 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,21)(19,23,27,26)(20,24,29,28)(25,31,32,30);(1,2)(3,8)(4,6)(5,7)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(10,13)(11,16)(14,15)(17,20,22,29)(18,24,21,28)(19,25,27,32)(23,30,26,31);(2,6)(5,8)(10,14)(13,15)(17,21,22,18)(19,26,27,23)(20,28,29,24)(25,31,32,30)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,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, 82109);