#312550

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

609 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=hi,c^2=hi,d^2=efh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh

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, 219746);