#495701

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#447
prime radical	#33530	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402897);