#478578

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=efi,c^2=ei,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385774);