#305172

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ehi,c^2=ef,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212368);