#149367

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dehi,c^2=deh,d^2=hi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=eh,g^c=gh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56563);