#338349

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,25)(21,23)(22,27)(24,29)(28,31)(30,32);(1,3,5,2)(4,7,8,6)(10,13)(11,14)(15,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,31)(25,26)(29,32);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(18,24)(19,27)(21,29)(22,30)(23,25)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 245545);