#244069

order	512
nilpotency class	3
isoclinism	isoclinic class of #233521
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224711, #72224723, #72224725
maximal normal subgroup of	#72224711, #72224723, #72224725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151265);