#307872

order	512
nilpotency class	3
isoclinism	isoclinic class of #264555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

623 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)(3,7)(4,5)(6,8)(17,18,24,29)(19,26,30,32)(20,25,21,31)(22,28,23,27);(2,5)(6,7)(9,10,13,16)(11,15,12,14)(17,19,23,21,24,30,22,20)(18,25,28,32,29,31,27,26);(10,14)(11,12)(15,16)(17,20,22,30,24,21,23,19)(18,26,27,31,29,32,28,25);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,16,14)(17,21,23,30,24,20,22,19)(18,26,28,25,29,32,27,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fh,d^2=ehi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=dhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 215068);