#362711

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2698 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)(3,4)(5,6)(7,8)(17,18,21,25,22,26,19,23)(20,28,27,24,30,31,29,32);(2,4)(9,10,12,14)(11,15,16,13)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,32,28);(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(2,4)(17,19,22,21)(18,25,26,23)(20,27,30,29)(24,28,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269907);