#369350

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1728 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,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, 276546);